Q.4 Using LMVT prove that : (a) tan > in 0,, (b) sin < for > 0 Q.5 Prove that if f is differentiable on [a, b] and if f (a) = f (b) = 0 then for any real there is an (a, b) such that f () + f ' () = 0. 0 Q.6 For what value of a, m and b does the function f () = a 0 m b satisfy the hypothesis of the mean value theorem for the interval [0, ]. Q.7 Suppose that on the interval [, 4] the function f is differentiable, f ( ) = and f ' () 5. Find the bounding functions of f on [, 4], using LMVT. Q.8 Let f, g be differentiable on R and suppose that f (0) = g (0) and f ' () g ' () for all 0. Show that f () g () for all 0. Q.9 Let f be continuous on [a, b] and differentiable on (a, b). If f (a) = a and f (b) = b, show that there eist distinct c, c in (a, b) such that f ' (c ) + f '(c ) =. Q.0 Let f () and g () be differentiable functions such that f ' () g () f () g ' () for any real. Show that between any two real solutions of f () = 0, there is at least one real solution of g () = 0. Q. Let f defined on [0, ] be a twice differentiable function such that, f " () for all [0, ] If f (0) = f (), then show that, f ' () < for all [0, ] Q. f () and g () are differentiable functions for 0 such that f (0) = 5, g (0) = 0, f () = 8, g () =. Show that there eists a number c satisfying 0 < c < and f ' (c) = g' (c). Q. If f,, are continuous in [a, b] and derivable in ]a, b[ then show that there is a value of c lying between a & b such that, f(a) f(b) f (c) (a) (b) (c) = 0 (a) (b) (c) Q.4 Show that eactly two real values of satisfy the equation = sin + cos. Q.5 Let a > 0 and f be continuous in [ a, a]. Suppose that f ' () eists and f ' () for all ( a, a). If f (a) = a and f ( a) = a, show that f (0) = 0. Q.6 Let a, b, c be three real number such that a < b < c, f () is continuous in [a, c] and differentiable in (a, c). Also f ' () is strictly increasing in (a, c). Prove that (c b) f (a) + (b a) f (c) > (c a) f (b) Q.7 Use the mean value theorem to prove, < ln <, > Q.8 Use mean value theorem to evaluate, Lim. Q.9 Using L.M.V.T. or otherwise prove that difference of square root of two consecutive natural numbers greater than N is less than. N Q.0 Prove the inequality e > ( + ) using LMVT for all R 0 and use it to determine which of the two numbers e and e is greater. EXERCISE 8 Q. If f () = sin & g () = tan, where 0 <, then in this interval : (A) both f () & g () are increasing functions (B) both f () & g () are decreasing functions (C) f () is an increasing function (D) g () is an increasing function [ JEE '97 (Scr), ] Q. Let a + b = 4, where a < and let g () be a differentiable function. If dg > 0 for all, prove that d a 0 b g( ) d g( ) d increases as (b a) increases. [JEE 97, 5] 0 Q.(a) Let h() = f() (f()) + (f()) for every real number. Then : (A) h is increasing whenever f is increasing (B) h is increasing whenever f is decreasing (C) h is decreasing whenever f is decreasing (D) nothing can be said in general. (b) f() =, for every real number, then the minimum value of f : (A) does not eist because f is unbounded (B) is not attained even though f is bounded (C) is equal to (D) is equal to. [ JEE '98, + ] Q.4(a) For all (0, ) : (A) e < + (B) log e ( + ) < (C) sin > (D) log e > (b) Consider the following statements S and R : S : Both sin & cos are decreasing functions in the interval (/, ). R : If a differentiable function decreases in an interval (a, b), then its derivative also decreases in (a, b). Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page of 5
Which of the following is true? (A) both S and R are wrong (B) both S and R are correct, but R is not the correct eplanation for S (C) S is correct and R is the correct eplanation for S (D) S is correct and R is wrong. (c) Let f () = e ( ) ( ) d then f decreases in the interval : (A) (, ) (B) (, ) (C) (, ) (D) (, + ) [JEE 000 (Scr.) ++ out of 5] Q.5(a) If f () = e ( ), then f() is (A) increasing on, (B) decreasing on R (C) increasing on R (D) decreasing on, (b) Let < p <. Show that the equation 4 p = 0 has a unique root in the interval, and identify it. [ JEE 00, + 5 ] Q.6 The length of a longest interval in which the function sin 4sin is increasing, is (A) (B) (C) (D) [JEE 00 (Screening), ] Q.7(a) Using the relation ( cos) <, 0 or otherwise, prove that sin (tan) >, 0,. 4 (b) Let f : [0, 4] R be a differentiable function. (i) Show that there eist a, b [0, 4], (f (4)) (f (0)) = 8 f (a) f (b) (ii) Show that there eist, with 0 < < < such that 4 0 f(t) dt = ( f ( ) + f ( ) ) [JEE 00 (Mains), 4 + 4 out of 60] ln, 0 Q.8(a) Let f () =. Rolle s theorem is applicable to f for [0, ], if = 0, 0 (A) (B) (C) 0 (D) (b) If f is a strictly increasing function, then Lim 0 f ( ) f () f () f (0) is equal to (A) 0 (B) (C) (D) [JEE 004 (Scr)] Q.9 If p () = 5 0 00 45 + 05, using Rolle's theorem, prove that at least one root of p() lies between (45 /00, 46). [JEE 004, out of 60] Q.0(a) If f () is a twice differentiable function and given that f() =, f() = 4, f() = 9, then (A) f '' () =, for (, ) (B) f '' () = f ' () =, for some (, ) (C) f '' () =, for (, ) (D) f '' () =, for some (, ) [JEE 005 (Scr), ] (b) f () is differentiable function and g () is a double differentiable function such that f () and f '() = g (). If f (0) +g (0) = 9. Prove that there eists some c (, ) such that g (c) g"(c)<0. [JEE 005 (Mains), 6] EXERCISE 9 Only one correct options. The function is monotonically decreasing for (A) (, ) (B) (0, ) (C) (0, ) (, ) (D) (, ). The values of p for which the function f() = p 4 p 5 + ln 5 decreases for all real is (A) (, ) (B) 4, (, ) 5 7 (C), (, ) (D) [, ) Successful People Replace the words like; "wish", "try" & "should" with "I Will". Ineffective People don't. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page of 5
. The set of all for which n ( + ) is equal to (A) > 0 (B) > (C) < < 0 (D) null set 4. Let f() = + a + b + 5 sin be an increasing function in the set of real numbers R. Then a & b satisfy the condition: (A) a b 5 > 0 (B) a b + 5 < 0 (C) a b 5 < 0 (D) a > 0 & b > 0 5. If f() = a sgn a sgn a ; g() = a for a >, a and R, where { } & [ ] denote the fractional part and integral part functions respectively, then which of the following statements holds good for the function h(), where (n a) h() = (n f() + n g()). (A) h is even and increasing (B) h is odd and decreasing (C) h is even and decreasing (D) h is odd and increasing 6. If f : [, 0] [, 0] is an non-decreasing function and g : [, 0] [, 0] is a non-increasing function. Let h() = f(g()) with h() =, then h() (A) lies in (, ) (B) is more than (C) is equal to (D) is not defined 7. Let f be a differentiable function of, R. If f() = 4 and f() [, 6], then minimum value of f(6) is (A) 6 (B) (C) 4 (D) none of these 8. For what values of a does the curve f() = (a a ) + cos is always strictly monotonic R. (A) a R (B) a < (C) < a < + (D) a < 9. If f() = ; g() = where 0 < <, then cos 6 6sin (A) both 'f' and 'g' are increasing functions (B) 'f' is decreasing & 'g' is increasing function (C) 'f' is increasing & 'g' is decreasing function (D) both 'f' & 'g' are decreasing function One or more than one correct options : 0. The set of values of a for which the function f() = + a + is an increasing function on [, ] is and decreasing in [, ] is, then : (A) : a (, ) (B) : a (, 4) (C) : a (, 4] (D) ; a [, ). If f is an even function then (A) f increases on (a, b) (B) f cannot be monotonic (C) f need not increases on (a, b) (D) f has inverse. Let g() = f(/) + f( ) and f() < 0 in 0 then g() : (A) decreases in 0, (B) decreases, (C) increases in 0, (D) increases in,. On which of the following intervals, the function 00 + sin is strictly increasing (A) (, ) (B) [0, ] (C) [/, ] (D) [0, /] 4. The function y = ( ) : (A) is its own inverse (B) decreases for all values of (C) has a graph entirely above -ais (D) is bound for all. 5. Let f and g be two functions defined on an interval such that f() 0 and g() 0 for all and f is strictly decreasing on while g is strictly increasing on then (A) the product function fg is strictly increasing on (B) the product function fg is strictly decreasing on I (C) fog() is monotonically increasing on (D) fog () is monotonically decreasing on EXERCISE 0 ma (, ) 0. Let f() =. Draw the graph of f() and hence comment on the nature of monotonic min (, ) 0 behaviour at =, 0,. 0. Let f() =. Find possible values of a such that f() is monotonically increasing at = 0. a 0. Find the relation between the constants a, b, c & d so that the function, f() = [a sin + b cos]@[c sin + d cos ] is always increasing. 4. Find the intervals of monotonocity for the following functions : f() = 5. If p, q, r be real, locate the intervals in which, f() = pq pr (a) increase (b) decrease 6. Find the values of a for which the function f() = (a + ) a + 9a decreases for all real values of. p pq q qr pr qr r, Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page of 5
7. Prove that for 0 < <, the inequality, < n ( ) < ( ). 8. For 0, identify which is greater (sin + tan) Or () hence find 0. sin tan 9. sin Prove that f() = is monotonically decreasing function for 0,. Hence prove that (i) sin sin(sin ) for 0,, cosec < (ii) < 6 sin 0. A (0, ), B, are two points on the graph given by y = sin + cos. Prove that there eists a point P on the curve between A & B such that tangent at P is parallel to AB. Find the co-ordinates of P.. Using Rolle s theorem prove that the equation + p = 0 has at least one real root in the interval (, ).. Show that e = has one & only one root between 0 &.. Find the interval in which the following function is increasing or decreasing : 4sin cos f() = in [0, ]. cos sin for 0 4. Show that the derivative of the function f() = vanishes on an infinite set of points of the 0 for 0 interval (0, ). f(a) f(b) 5. Assume that f is continuous on [a, b] a > 0 and differentiable in (a, b) such that =. Prove that a b f(0 ) there eists 0 (a, b) such that f( 0 ) =. 6. Find the greatest & least value of f() = sin 7. Show that, + n 8. tan Prove the inequality, > tan 0 for all 0. for 0 < < <. n in,. 9. A function f is differentiable in the interval 0 5 such that f(0) = 4 & f(5) =. If g() = that there eists some c (0, 5) such that g() = 6 5. f(), then prove 0. Prove that for all R e + e ( + ) +.. Let f (sin) < 0 and f (sin ) > 0, 0, and g() = f(sin ) + f(cos ), then find the interval in which g() is increasing and decreasing.. If a + (b/) c for all positive where a > 0 and b > 0 then show that 7ab 4c.. Prove that for 0 p & for any a > 0, b > 0 the inequality (a + b) p a p + b p. 4. Find the greatest and the least values of the function f() defined as below. f() = minimum of {t 4 8t 6t + 4t ; t }, <. maimum of t sin t ; t, 4. 4 5. If a > b > 0, with the aid of Largrange's formula prove the validity of the inequailities nb n (a b) < a n b n < na n (a b), if n >. Also prove that the inequalities of the opposite sense if 0 < n <. MAXIMA - MINIMA FUNCTIONS OF A SINGLE VARIABLE HOW MAXIMA & MINIMA ARE CLASSIFIED. A function f() is said to have a maimum at = a if f(a) is greater than every other value assumed by f() in the immediate neighbourhood of = a. Symbolically f( a) f( a h) f( a) f( a h) = a gives maima for a sufficiently small positive h. Similarly, a function f() is said to have a minimum Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 4 of 5
value at = b if f(b) is least than every other value assumed by f() in the immediate neighbourhood at = b. Symbolically if f( b) f( b h) f( b) f( b h) = b gives minima for a sufficiently small positive h. Note that : (i) the maimum & minimum values of a function are also known as local/relative maima or local/relative minima as these are the greatest & least values of the function relative to some neighbourhood of the point in question. (ii) the term 'etremum' or (etremal) or 'turning value' is used both for maimum or a minimum value. (iii) a maimum (minimum) value of a function may not be the greatest (least) value in a finite interval. (iv) a function can have several maimum & minimum values & a minimum value may even be greater than a (v) maimum value. maimum & minimum values of a continuous function occur alternately & between two consecutive maimum values there is a minimum value & vice versa.. A NECESSARY CONDITION FOR MAXIMUM & MINIMUM : If f() is a maimum or minimum at = c & if f (c) eists then f (c) = 0. Note : (i) (ii) (iii) The set of values of for which f () = 0 are often called as stationary points or critical points. The rate of change of function is zero at a stationary point. In case f (c) does not eist f(c) may be a maimum or a minimum & in this case left hand and right hand derivatives are of opposite signs. The greatest (global maima) and the least (global minima) values of a function f in an interval [a, b] are f(a) or f(b) or are given by the values of for which f () = 0. Critical points are those where d y = 0, if it eists, or it fails to eist either by virtue of a vertical tangent or (iv) d by virtue of a geometrical sharp corner but not because of discontinuity of function.. SUFFICIENT CONDITION FOR EXTREME VALUES : f (c-h) > 0 f (c+ h) < 0 = c is a point of local maima, where f (c) = 0. h is a sufficiently small positive f (c-h) < 0 Similarly f (c+ h) > 0 = c is a point of local minima, where f (c) = 0. quantity Note : If f () does not change sign i.e. has the same sign in a certain complete neighbourhood of c, then f() is either strictly increasing or decreasing throughout this neighbourhood implying that f(c) is not an etreme value of f. 4. USE OF SECOND ORDER DERIVATIVE IN ASCERTAINING THE MAXIMA OR MINIMA: (a) f(c) is a minimum value of the function f, if f (c) = 0 & f (c) > 0. (b) f(c) is a maimum value of the function f, f (c) = 0 & f (c) < 0. Note : if f (c) = 0 then the test fails. Revert back to the first order derivative check for ascertaning the maima or minima. 5. SUMMARYWORKING RULE : FIRST : When possible, draw a figure to illustrate the problem & label those parts that are important in the problem. Constants & variables should be clearly distinguished. SECOND : Write an equation for the quantity that is to be maimised or minimised. If this quantity is denoted by y, it must be epressed in terms of a single independent variable. his may require some algebraic manipulations. THIRD : If y = f () is a quantity to be maimum or minimum, find those values of for which dy/d = f () = 0. FOURTH : Test each values of for which f () = 0 to determine whether it provides a maimum or minimum or neither. The usual tests are : (a) If d²y/d² is positive when dy/d = 0 y is minimum. If d²y/d² is negative when dy/d = 0 y is maimum. If d²y/d² = 0 when dy/d = 0, the test fails. (b) If d y positive for d is 0 zero for 0 a maimum occurs at = 0. negative for 0 But if dy/d changes sign from negative to zero to positive as advances through o there is a minimum. If dy/d does not change sign, neither a maimum nor a minimum. Such points are called INFLECTION POINTS. FIFTH : If the function y = f () is defined for only a limited range of values a b then eamine = a & = b for possible etreme values. SIXTH : If the derivative fails to eist at some point, eamine this point as possible maimum or minimum. Important Note : Given a fied point A(, y ) and a moving point P(, f ()) on the curve y = f(). Then AP will be maimum or minimum if it is normal to the curve at P. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 5 of 5
If the sum of two positive numbers and y is constant than their product is maimum if they are equal, i.e. + y = c, > 0, y > 0, then y = 4 [ ( + y) ( y) ] If the product of two positive numbers is constant then their sum is least if they are equal. i.e. ( + y) = ( y) + 4y 6. USEFUL FORMULAE OF MENSURATION TO REMEMBER : Volume of a cuboid = lbh. Surface area of a cuboid = (lb + bh + hl). Volume of a prism = area of the base height. Lateral surface of a prism = perimeter of the base height. Total surface of a prism = lateral surface + area of the base (Note that lateral surfaces of a prism are all rectangles). Volume of a pyramid = area of the base height. Curved surface of a pyramid = (perimeter of the base) slant height. (Note that slant surfaces of a pyramid are triangles). Volume of a cone = r h. Curved surface of a cylinder = rh. Total surface of a cylinder = rh + r. Volume of a sphere = 4 r. Surface area of a sphere = 4 r. Area of a circular sector = r, when is in radians. 7. SIGNIFICANCE OF THE SIGN OF ND ORDER DERIVATIVE AND POINTS OF INFLECTION : The sign of the nd order derivative determines the concavity of the curve. Such points such as C & E on the graph where the concavity of the curve changes are called the points of inflection. From the graph we find that if: d y (i) > 0 concave upwards d (ii) d y < 0 concave downwards. d At the point of inflection we find that d y = 0 & d d y changes sign. d Inflection points can also occur if d y fails to eist. For eample, consider the graph of the function defined d as, f () = [ / 5 for (, ) for (, ) Note that the graph ehibits two critical points one is a point of local maimum & the other a point of inflection. EXERCISE Q. A cubic f() vanishes at = & has relative minimum/maimum at = and =. 4 If f ()d =, find the cubic f (). Q. Investigate for maima & minima for the function, f () = [ (t ) (t ) + (t ) (t ) ] dt Q. Find the maimum & minimum value for the function ; (a) y = + sin, 0 (b) y = cos cos 4, 0 Q.4 Suppose f() is real valued polynomial function of degree 6 satisfying the following conditions ; Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 6 of 5
(a) f has minimum value at = 0 and (b) f has maimum value at = (c) f() for all, Limit 0 0 ln 0 0 =. Determine f (). Q.5 Find the maimum perimeter of a triangle on a given base a and having the given vertical angle. Q.6 The length of three sides of a trapezium are equal, each being 0 cms. Find the maimum area of such a trapezium. Q.7 The plan view of a swimming pool consists of a semicircle of radius r attached to a rectangle of length 'r' and width 's'. If the surface area A of the pool is fied, for what value of 'r' and 's' the perimeter 'P' of the pool is minimum. Q.8 For a given curved surface of a right circular cone when the volume is maimum, prove that the semi vertical angle is sin. 6 Q.9 Of all the lines tangent to the graph of the curve y =, find the equations of the tangent lines of minimum and maimum slope. Q.0 A statue 4 metres high sits on a column 5.6 metres high. How far from the column must a man, whose eye level is.6 metres from the ground, stand in order to have the most favourable view of statue. Q. By the post office regulations, the combined length & girth of a parcel must not eceed metre. Find the volume of the biggest cylindrical (right circular) packet that can be sent by the parcel post. Q. A running track of 440 ft. is to be laid out enclosing a football field, the shape of which is a rectangle with semi circle at each end. If the area of the rectangular portion is to be maimum, find the length of its sides. Use :. 7 Q. A window of fied perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted with clean glass. The clear glass transmits three times as much light per square meter as the coloured glass does. What is the ratio of the sides of the rectangle so that the window transmits the maimum light? Q.4 A closed rectangular bo with a square base is to be made to contain 000 cubic feet. The cost of the material per square foot for the bottom is 5 paise, for the top 5 paise and for the sides 0 paise. The labour charges for making the bo are Rs. /-. Find the dimensions of the bo when the cost is minimum. Q.5 Find the area of the largest rectangle with lower base on the -ais & upper vertices on the curve y =. Q.6 A trapezium ABCD is inscribed into a semicircle of radius l so that the base AD of the trapezium is a diameter and the vertices B & C lie on the circumference. Find the base angle of the trapezium ABCD which has the greatest perimeter. a b Q.7 If y = has a turning value at (, ) find a & b and show that the turning value is a maimum. ( ) ( 4) Q.8 Prove that among all triangles with a given perimeter, the equilateral triangle has the maimum area. Q.9 A sheet of poster has its area 8 m². The margin at the top & bottom are 75 cms and at the sides 50 cms. What are the dimensions of the poster if the area of the printed space is maimum? Q.0 A perpendicular is drawn from the centre to a tangent to an ellipse + y =. Find the greatest value of a b the intercept between the point of contact and the foot of the perpendicular. Q. Consider the function, F () = ( t t) dt, R. (a) Find the and y intercept of F if they eist. (b) Derivatives F ' () and F '' (). (c) The intervals on which F is an increasing and the invervals on which F is decreasing. (d) Relative maimum and minimum points. (e) Any inflection point. Q. A beam of rectangular cross section must be sawn from a round log of diameter d. What should the width and height y of the cross section be for the beam to offer the greatest resistance (a) to compression; (b) to bending. Assume that the compressive strength of a beam is proportional to the area of the cross section and the bending strength is proportional to the product of the width of section by the square of its height. Q. What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle of base 6 ft. & altitude ft? Assume that one side of the rectangle lies on the base of the triangle. Q.4 The flower bed is to be in the shape of a circular sector of radius r & central angle. If the area is fied & perimeter is minimum, find r and. Q.5 The circle + y = cuts the -ais at P & Q. Another circle with centre at Q and varable radius intersects the first circle at R above the -ais & the line segment PQ at S. Find the maimum area of the triangle QSR. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 7 of 5
EXERCISE Q. The mass of a cell culture at time t is given by, M (t) = t 4e (a) Find Lim M(t) and Lim M(t) (b) Show that dm = M( M ) t t dt (c) Find the maimum rate of growth of M and also the vlaue of t at which occurs. Q. Find the cosine of the angle at the verte of an isosceles triangle having the greatest area for the given constant length l of the median drawn to its lateral side. Q. From a fied point A on the circumference of a circle of radius 'a', let the perpendicular AY fall on the tangent at a point P on the circle, prove that the greatest area which the APY can have a is sq. units. 8 Q.4 Given two points A (, 0) & B (0, 4) and a line y =. Find the co-ordinates of a point M on this line so that the perimeter of the AMB is least. Q.5 A given quantity of metal is to be casted into a half cylinder i.e. with a rectangular base and semicircular ends. Show that in order that total surface area may be minimum, the ratio of the height of the cylinder to the diameter of the semi circular ends is /( + ). Q.6 Depending on the values of p R, find the value of 'a' for which the equation + p + p = a has three distinct real roots. Q.7 Show that for each a > 0 the function e a. a² has a maimum value say F (a), and that F () has a minimum value, e e/. Q.8 For a > 0, find the minimum value of the integral ( a 4 a )e d. Q.9 Find the maimum value of the integral a e d where a. ln when 0 Q.0 Consider the function f () = 0 for 0 (a) Find whether f is continuous at = 0 or not. (b) Find the minima and maima if they eist. (c) Does f ' (0)? Find Lim f '(). 0 (d) Find the inflection points of the graph of y = f ().. Q. Consider the function y = f () = ln ( + sin ) with. Find (a) the zeroes of f () (b) inflection points if any on the graph (c) local maima and minima of f () (d) asymptotes of the graph (e) sketch the graph of f () and compute the value of the definite integral f ()d. Q. A right circular cone is to be circumscribed about a sphere of a given radius. Find the ratio of the altitude of the cone to the radius of the sphere, if the cone is of least possible volume. Q. Find the point on the curve 4 ² + a²y² = 4 a², 4 < a² < 8 that is farthest from the point (0, ). 5 Q.4 Find the set of value of m for the cubic + = log ) 4 ( m has distinct solutions. Q.5 Let A (p, p), B (q, q), C (r, r) be the vertices of the triangle ABC. A parallelogram AFDE is drawn with vertices D, E & F on the line segments BC, CA & AB respectively. Using calculus, show that maimum area of such a parallelogram is : 4 (p + q) (q + r) (p r). Q.6 A cylinder is obtained by revolving a rectangle about the ais, the base of the rectangle lying on the ais and the entire rectangle lying in the region between the curve y = & the ais. Find the maimum possible volume of the cylinder. Q.7 For what values of a does the function f () = + (a 7) + (a 9) have a positive point of maimum. Q.8 Among all regular triangular prism with volume V, find the prism with the least sum of lengths of all edges. How long is the side of the base of that prism? Q.9 A segment of a line with its etremities on AB and AC bisects a triangle ABC with sides a, b, c into two equal areas. Find the length of the shortest segment. Q.0 What is the radius of the smallest circular disk large enough to cover every acute isosceles triangle of a given perimeter L? Q. Find the magnitude of the verte angle of an isosceles triangle of the given area A such that the radius r of the circle inscribed into the triangle is the maimum. Q. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6 r. a 0 5 a Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 8 of 5
Q. The function f () defined for all real numbers has the following properties (i) f (0) = 0, f () = and f ' () = k( )e for some constant k > 0. Find (a) the intervals on which f is increasing and decreasing and any local maimum or minimum values. (b) the intervals on which the graph f is concave down and concave up. (c) the function f () and plot its graph. Q.4 Find the minimum value of sin + cos + tan + cot + sec + cosec for all real. Q.5 Use calculus to prove the inequality, sin in 0. You may use the inequality to prove that, cos in 0 EXERCISE Q. A conical vessel is to be prepared out of a circular sheet of gold of unit radius. How much sectorial area is to be removed from the sheet so that the vessel has maimum volume. [ REE '97, 6 ] Q.(a) The number of values of where the function f() = cos + cos attains its maimum is : (A) 0 (B) (C) (D) infinite (b) Suppose f() is a function satisfying the following conditions : (i) f(0) =, f() = (ii) f has a minimum value at = 5 and a a a b (iii) for all f () = b b (a b) a b a b Where a, b are some constants. Determine the constants a, b & the function f(). Q. Find the points on the curve a + by + ay = c ; c > b > a > 0, whose distance from the origin is minimum. [ REE '98, 6] Q.4 The function f() = t (e t ) (t ) (t ) (t ) 5 dt has a local minimum at = (A) 0 (B) (C) (D) Q.5 Find the co-ordinates of all the points P on the ellipse ( /a ) + (y /b ) = for which the area of the triangle PON is maimum, where O denotes the origin and N the foot of the perpendicular from O to the tangent at P. [JEE '99, 0 out of 00] Q.6 Find the normals to the ellipse ( /9) + (y /4) = which are farthest from its centre. [REE '99, 6] Q.7 Find the point on the straight line, y = + which is nearest to the circle, 6 ( + y ) + 8 y 50 = 0. [REE 000 Mains, out of 00] Q.8 Let f () = [ for 0. Then at = 0, ' f ' has : for 0 (A) a local maimum (B) no local maimum (C) a local minimum (D) no etremum. Q.9 Find the area of the right angled triangle of least area that can be drawn so as to circumscribe a rectangle of sides 'a' and 'b', the right angles of the triangle coinciding with one of the angles of the rectangle. Q.0(a) Let f() = ( + b ) + b + and let m(b) be the minimum value of f(). As b varies, the range of m (b) is (A) [0, ] (B) 0, (C), (D) (0, ] (b) The maimum value of (cos ) (cos )... (cos n ), under the restrictions O <,,..., n < and cot cot... cot n = is (A) (B) (C) (D) n/ n n Q.(a) If a, a,..., a n are positive real numbers whose product is a fied number e, the minimum value of a + a + a +... + a n + a n is (A) n(e) /n (B) (n+)e /n (C) ne /n (D) (n+)(e) /n (b) A straight line L with negative slope passes through the point (8,) and cuts the positive coordinates aes at points P and Q. Find the absolute minimum value of OP + OQ, as L varies, where O is the origin. Q.(a) Find a point on the curve + y = 6 whose distance from the line + y = 7, is minimum. (b) For a circle + y = r, find the value of r for which the area enclosed by the tangents drawn from the point P(6, 8) to the circle and the chord of contact is maimum. [JEE-0, Mains- out of 60] Q.(a) Let f () = + b + c + d, 0 < b < c. Then f (A) is bounded (B) has a local maima (C) has a local minima (D) is strictly increasing [JEE 004 (Scr.)] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 9 of 5
( ) (b) Prove that sin 0,. (Justify the inequality, if any used). Q.4 If P() be a polynomial of degree satisfying P( ) = 0, P() = 6 and P() has maimum at = and P'() has minima at =. Find the distance between the local maimum and local minimum of the curve. [JEE 005 (Mains), 4 out of 60] Q.5(a) If f () is cubic polynomial which has local maimum at =. If f () = 8, f () = and f '() has local maima at = 0, then (A) the distance between (, ) and (a, f (a)), where = a is the point of local minima is 5. (B) f () is increasing for [, 5 ] (C) f () has local minima at = (D) the value of f(0) = 5 e 0 (b) f () = e and g () = f tdt, [, ] then g() has e 0 (A) local maima at = + ln and local minima at = e (B) local maima at = and local minima at = (C) no local maima (D) no local minima (c) If f () is twice differentiable function such that f (a) = 0, f (b) =, f (c) =, f (d) =, f (e) = 0, where a < b < c < d < e, then find the minimum number of zeros of g() f '() f ().f"() interval [a, e]. [JEE 006, 6] EXERCISE 4 Only one correct option. The greatest value of f() = ( + ) / ( ) / on [0, ] is: (A) (B) (C) (D) /. The function 'f' is defined by f() = p ( ) q for all R, where p,q are positive integers, has a maimum value, for equal to: pq p (A) (B) (C) 0 (D) pq pq. The coordinates of the point on the curve = 4y, which is at least distance from the line y = 4 is (A) (, ) (B) (, ) (C) (, ) (D) none 4. Tangents are drawn to + y = 6 from the point P(0, h). These tangents meet the -ais at A and B. If the area of triangle PAB is minimum, then 5. (A) h = (B) h = 6 (C) h = 8 (D) h= 4 A function f is such that f() = f() = 0 and f has a local maimum of 7 at =, then f() may be (A) f() = 7 ( ) n n N n 4 (B) f() = 7 ( ) n n (C) f() = 7 + ( ) n n (D) f() 7 ( ) n n 4 tan, 4 6. f() =, then, 4 (A) f() has no point of local maima (B) f() has only one point of local maima (C) f() has eactly two points of local maima (D) f() has eactly two points of local minimas 05, the set of values of b for which f() have greatest value at = is given 7. Let f() = log b, by : (A) b (B) b = {, } 8. (C) b (, ) (D) 0, U, 0 A tangent to the curve y = is drawn so that the abscissa 0 of the point of tangency belongs to the interval (0, ]. The tangent at 0 meets the ais and yais at A & B respectively. The minimum area of the triangle OAB, where O is the origin is 9. (A) (B) 4 (C) (D) none 9 9 9 The lower corner of a leaf in a book is folded over so as to just reach the inner edge of the page. The fraction of width folded over if the area of the folded part is minimum is: (A) 5/8 (B) / (C) /4 (D) 4/5 0. n {a, a,..., a 4,...} is a progression where a n =. The largest term of this progression is: n 00 (A) a 6 (B) a 7 (C) a 8 (D) none. (sin tan ) If f() = + then the range of f() is in the Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 40 of 5
5 7 7 (A), (B) 4 4 0, (C) 4, (D) 4 4, 4 4 {} {}. Let f() = sin + cos where a > 0 and {. } denotes the fractional part function. Then the set of a a values of a for which f can attain its maimum values is 4 4 (A) 0, (B), (C) (0, ) (D) none of these. A and B are the points (, 0) and (0, ) respectively. The coordinates of the point P on the line + y + = 0 are (A) (7, 5) if PA PB is maimum (B), if PA PB is maimum 5 5 (C) (7, 5) if PA PB is minimum (D), if PA PB is minimum 5 5 4. The maimum area of the rectangle whose sides pass through the angular points of a given rectangle of sides a and b is (A) (ab) (B) (a + b) (C) (a + b ) (D) none of these 5. Number of solution(s) satisfying the equation, = log ( + ) log is: (A) (B) (C) (D) none 6. Least value of the function, f() = + is: (A) 0 (B) / (C) / (D) 7. A straight line through the point (h, k) where h > 0 and k > 0, makes positive intercepts on the coordinate aes. Then the minimum length of the line intercepted between the coordinate aes is (A) (h / + k / ) / (B) (h / + k / ) / (C) (h / k / ) / (D) (h / k / ) / 8. The value of a for which the function f() = (4a ) ( + log 5) + (a 7) cot sin does not posses critical points is (A) (, 4/) (B) (, ) (C) [, ) (D) (, ) 9. The minimum value of n sin n is cos (A) (B) (C) ( + n/ ) (D) None of these 0. The altitude of a right circular cone of minimum volume circumscribed about a sphere of radius r is (A) r (B) r (C) 5 r (D) none of these One or more than one correct options. Let f() = 40/( 4 + 8 8 + 60), consider the following statement about f(). (A) f() has local minima at = 0 (B) f() has local maima at = 0 (C) absolute maimum value of f() is not defined (D) f() is local maima at =, =. Maimum and minimum values of the function, f() = cos ( + ) + sin ( + ) 0 < < 4 occur at : (A) = (B) = (C) = (D) =. If lim a f() = lim a function, then [f()] ( [. ] denotes the greater integer function) and f() is non-constant continuous (A) lim a f() is integer (B) lim f() is non-integer a (C) f () has local maimum at = a (D) f () has local minima at = a 4. If the derivative of an odd cubic polynomial vanishes at two different values of then (A) coefficient of & in the polynomial must be same in sign (B) coefficient of & in the polynomial must be different in sign (C) the values of where derivative vanishes are closer to origin as compared to the respective roots on either side of origin. (D) the values of where derivative vanishes are far from origin as compared to the respective roots on either side of origin. 5. Let f() = ln ( ) + sin. Then (A) graph of f is symmetrical about the line = (B)graph of f is symmetrical about the line = (C) maimum value of f is (D) minimum value of f does not eist 6. The curve y = has: (A) =, the point of inflection (B) = +, the point of inflection (C) =, the point of minimum (D) =, the point of inflection 7. If the function y = f () is represented as, = (t) = t 5 t 0 y = (t) = 4 t + 7 t t 8 t + ( < t < ), (A) y then: ma = (B) y ma = 4 (C) y min = 67/4 (D) y min = 69/4 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 4 of 5
a b c 8. The maimum and minimum values of y = are those for which A B C (A) a + b + c y (A + B + C) is equal to zero (B) a + b + c y (A + B + C) is a perfect square dy d y (C) = 0 and d 0 d (D) a + b + c y (A + B + C) is not a perfect square 9. f() is cubic polynomial which has local maimum at =, If f() = 8, f() = and f() has local minima at = 0, then [IIT 006, (5, )] (A) the distance between point of maima and minima is 5. (B) f() is increasing for [, 5 ) (C) f() has local minima at = (D) the value of f(0) = 5 EXERCISE 5. Find the area of the largest rectangle with lower base on the -ais & upper vertices on the curve y = ².. Find the cosine of the angle at the verte of an isosceles triangle having the greatest area for the given constant length of the median drawn to its lateral side. a. Find the set of value(s) of 'a' for which the function f () = + (a + ) + (a ) + possess a negative point of inflection. 4. The fuel charges for running a train are proportional to the square of the speed generated in m.p.h. & costs Rs. 48/- per hour at 6 mph. What is the most economical speed if the fied charges i.e. salaries etc. amount to Rs. 00/- per hour. 5. The three sides of a trapezium are equal each being 6 cms long, find the area of the trapezium when it is maimum. 6. What are the dimensions of the rectangular plot of the greatest area which can be laid out within a triangle of base 6 ft. & altitude ft? Assume that one side of the rectangle lies on the base of the triangle. 7. A closed rectangular bo with a square base is to be made to contain 000 cubic feet. The cost of the material per square foot for the bottom is 5 paise, for the top 5 paise and for the sides 0 paise. The labour charges for making the bo are Rs. /-. Find the dimensions of the bo when the cost is minimum. 8. Find the point on the curve, 4 + a y = 4 a, 4 < a < 8, that is farthest from the point (0, ). 9. A cone is circumscribed about a sphere of radius ' r '. Show that the volume of the cone is minimum when its semi vertical angle is, sin. 0. Find the values of 'a' for which the function f() = a + (a + ) + (a ) + possess a negative point of minimum.. A figure is bounded by the curves, y = +, y = 0, = 0 & =. At what point (a, b), a tangent should be drawn to the curve, y = + for it to cut off a trapezium of the greatest area from the figure.. Prove that the least perimeter of an isosceles triangle in which a circle of radius ' r ' can be inscribed is 6 r.. Find the polynomial f () of degree 6, which satisfies Limit f ( ) 0 = e and has local maimum at = and local minimum at = 0 &. 4. Two towns located on the same side of the river agree to construct a pumping station and filteration plant at the river s edge, to be used jointly to supply the towns with water. If the distance of the two towns from the river are a & b and the distance between them is c, show that the pipe lines joining them to the pumping station is atleast as great as c 4ab. 5. Find the co-ordinates of all the points P on the ellipse ( /a ) + (y /b ) = for which the area of the triangle PON is maimum, where O denotes the origin and N the foot of the perpendicular from O to the tangent at P. [IIT 999, 0] 6. If p() be a polynomial of degree satisfying p( ) = 0, p() = 6 and p() has maima at = and p() has minima at =. Find the distance between the local maima and local minima of the curve. [IIT 005] / Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 4 of 5
Q. y = TANGENT & NORMAL EXERCISE - or + y = Q. = when t =, m ; 5 4y = if t, m = /] Q. (0, ) Q.7 T : y = 0 ; N : + y = 0 Q.8 + y = / & + y = / Q.9 (a) n = Q. a = Q.4 ; 4y = Q.6 a = / ; b = /4 ; c = Q.0 e / Q. (b) a b = a b Q. = tan C Q.5 EXERCISE - Q. /9 m/min Q. (i) 6 km/h (ii) km/hr Q. (4, ) & ( 4, /) Q.4 /8 cm/min Q.5 + 6 cu. cm/sec Q.6 /48 cm/s Q.7 0.05 cm/sec Q.8 4 cm/s Q.9 00 r / (r + 5)² km² / h Q.0 66 7 5 Q. (a) m/min., (b) 4 88 Q. = tan 4 7 m m Q. 4 cm/sec. m/min. Q.4 (a) r = ( + t) /4, (b) t = 80 Q.5 (a) 5.0, (b) 80 7 EXERCISE - Q. + y = 0 or y = 0 Q. D Q.4 D Q.5 D EXERCISE - 4. B. B. D 4. B 5. B 6. A 7. B 8. A 9. B 0. B. B. B. A 4. C 5. B 6. ABD 7. AC 8. AB 9. ABC 0. AC. CD. BD EXERCISE - 5. a =, b =, c = 0. (9/4, /8). 5. (i) 8b 7 at (0, 0); tan at (8, 6), (8, 6) (ii) / (iii) tan 4 at 7 6. (i) cm/min (ii) cm /min. 7. (4, ) & ( 4, /). + y = 0, = y. ± 7. 5y + 4 = 4 y 9. t = k H c 4. y = 5 6. a, 4 0. y = MONOTONOCITY EXERCISE - 6 Q. (a) I in (, ) & D in (, ) (b) I in (, ) & D in (, 0) (0, ) (c) I in (0, ) & D in (, ) (, ),,, (d) I for > or < < 0 & D for < or 0 < < Q. (, 0) (, ) Q. (a) I in [0, /4) (7/4, ] & D in (/4, 7 /4) (b) I in [0, /6) (/, 5/6) (/, ] & D in (/6, /) (5/6, /)] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 4 of 5
Q.5 continuous but not diff. at = Q.6 a < 5 or a > 5 Q.7 (a) (/6)+(/)ln, (/) (/)ln, (b) least value is equal to (/e) /e, no greatest value, (c) & 0 Q.8 [, ) Q.0 a (, ] [, ) Q. [ 7, ) [, ] Q. increasing in (/, /) & decreasing in [0, /) (/, ] Q. 0 a Q.4 in (, ) and in (, ) Q.5 (6, ) Q.6 a 0 5 Q.7 (a) (, 0] ; (b) in, 5 5 and in (,), { } ; (c) = ; (d) removable discont. at = (missing point) and non removable discont. at = (infinite type) (e) Q.4 (, 0) (, ) Q.5 (b a) /4 EXERCISE - 7 mb na Q. c = which lies between a & b Q.6 a =, b = 4 and m = m n Q.7 y = 5 9 and y = 5 + Q.8 0 EXERCISE - 8 Q. C Q. (a) A, C ; (b) D Q.4 (a) B ; (b) D ; (c) C Q.5 (a) A, (b) cos cos p Q.6 A Q.8 (a) D ; (b) C Q.0 (a) D EXERCISE - 9. C. B. B 4. C 5. D 6. C 7. A 8. C 9. C 0. CD. BC. BC. ABD 4. AB 5. ABD EXERCISE - 0. Neither increasing nor decreasing at =, increasing at = 0,.. a 0. ad > bc 4. in (, ] [, ) & D in [, ] 5. (a) < (p + q + r ), > 0 (b) ( (p + q + r ), 0) 6. (, ] 8. sin + tan >, limit = 0 0.,. increasing on [0, /] and decreasing on [/, ] 6 6. (/6) + (/) n, (/) (/) n. increasing when,, decreasing when 0,. 4 4. Prove that for 0 p & for any a > 0, b > 0 the inequality (a + b) p a p + b p. 4. greatest = 4, least = 8 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MAXIMA - MINIMA EXERCISE - Q. f () = + + Q. ma. at = ; f() = 0, min. at = 7/5 ; f(7/5) = 08/5 Q. (a) Ma at =, Ma value =, Min. at = 0, Min value = 0 (b) Ma at = /6 & also at = 5 /6 and Ma value = /, Min at = /, Min value = Q.4 f () = 6 5 5 + 4 Q.5 P ma = a cosec Q.6 75 sq. units Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 44 of 5
Q.7 r = A 4 s = A 4 Q.9 + 4y 9 = 0 ; 4y + 9 = 0 Q.0 4 m Q. / cu m Q. 0 ', 70 ' Q. 6/(6 + ) Q.4 side 0', height 0' Q.5 sq. units Q.6 = 60 0 Q.7 a =, b = 0 Q.9 width m, length m Q.0 a b Q. (a) (, 0), (0, 5/6) ; (b) F ' () = ( ), F '' () =, (c) increasing (, 0) (, ), decreasing (0, ) ; (d) (0, 5/6) ; (, /) ; (e) = / Q. (a) = y = d, (b) = d, y = d Q. 6' 8' Q.4 r = A, = radians Q.5 4 EXERCISE - Q. (a) 0,, (c) 4, t = ln 4 Q. cos A = 0.8 Q.4 (0, 0) Q.6 p < a < p p + p if p > 0 ; + p < a < p if p < 0 Q.8 4 when a = 7 7 Q.9 Maimum value is (e + e ) when a = Q.0 (a) f is continuous at = 0 ; (b) e ; (c) does not eist, does not eist ; (d) pt. of inflection = Q. (a) =,, 0,,, (b) no inflection point, (c) maima at = and and no minima, (d) = and =, (e) ln Q. 4 Q. (0, ) & ma. distance = 4 Q.4 m, 6 4V ( c a b)(a b c) Q.7 (, ) (, 9/7) Q.8 H = = Q.9 Q.0 L/4 Q. Q. (a) increasing in (0, ) and decreasing in (, 0) (, ), local min. value = 0 and local ma. value = / Q.6 (b) concave up for (, ) ( +, ) and concave down in ( ), ( + ) (c) f () = e Q.4 EXERCISE - Q. sq. units Q. (a) B, (b) a = 4 ; b = 5 4 ; f() = 4 ( 5 + 8) Q. c c ( a b), ( a b) & c c ( a b), ( a b) Q.4 (a) B, D, 4 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 45 of 5 Q.5 ± a a b, ± a b b Q.6 ± ± y = 5 Q.7 ( 9/, )
Q.8 A Q.9 ab Q.0 (a) D ; (b) A Q. (a) A ; (b) 8 Q. (a) (, ) ; (b) 5 Q. (a) D Q.4 4 65 Q.5 (a) B, C; (b) A, B, (c) 6 solutions EXERCISE - 4. B. D. A 4. D 5. A 6. C 7. D 8. B 9. B 0. B. B. A. A 4. B 5. A 6. D 7. A 8. A 9. C 0. D. ACD. AC. AD 4. BC 5. ACD 6. ABD 7. BD 8. BC 9. BC EXERCISE - 5. sq. units. cos A = 0.8. (, ) (0, ) 4. 40 mph 5. 7 sq. cms 6. 6 8 7. side 0', height 0' 8. (0, ) 0. (, ). 5. ± a a b, ± a b b 5, 4. f () = 4 5 5 + 6 6. 4 65 For 9 Years Que. from IIT-JEE(Advanced) & 5 Years Que. from AIEEE (JEE Main) we distributed a book in class room Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone :0 90 90 7779, 0 9890 5888. page 46 of 5