08 - DIFFERENTIAL CALCULUS Page 1 ( Answers at the end of all questions ) ( d ) it is at a constant distance from the o igin [ AIEEE 2005 ]

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Terence Willis
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1 08 - DIFFERENTIAL CALCULUS Page ( ) sec sec... sec n n n n n n sec ( b ) cosec ( c ) tan ( d ) tan [ AIEEE 005 ] ( ) The normal to the curve = a ( cos θ + θ sin θ ), y = a ( sin θ θ cos θ ) at any point θ such that it passes through the origin π ( b ) it makes angle + θ with the X-a ( c ) it passes through ( a π, - a ) ( d ) it at a constant dtance from the o igin [ AIEEE 005 ] ( ) A function matched below agains an interval where it supposed to be increasing. Which of the following pairs correctly matched? Interval Fu ction Interval Function ( -, ) + + ( b ) [, ) ( c ) ( -, ) - + ( d ) ( -, - 4 ) ( 4 ) Let α and β be the dtinct roots of the equation a + b + c = 0. Then cos ( a - + b + c ) im equal to α ( - α ) a ( α - β ) a ( b ) 0 ( c ) - ( α - β ) ( d ) [ AIEEE 005 ] ( α - β ) [ AIEEE 005 ] ( 5 ) Suppose f ( ) differentiable at = and f ( + h ) h 0 h = 5, then f ( ) equals ( b ) 4 ( c ) 5 ( d ) 6 [ AIEEE 005 ]

2 08 - DIFFERENTIAL CALCULUS Page ( 6 ) Let f be differentiable for al. If f ( ) = - and f ( ) for [, 6 ], then f ( 6 ) 8 ( b ) f ( 6 ) < 8 ( c ) f ( 6 ) < 5 ( d ) f ( 6 ) = 5 [ AIEEE 005 ] ( 7 ) If f a real valued differentiable function satfying l f ( ) - f ( y ) l ( y ),, y R and f ( 0 ) = 0, then f ( ) equals - ( b ) 0 ( c ) ( d ) [ AIEEE 005 ] ( 8 ) A spherical iron ball 0 cm in radius coated with laye of ice of uniform thickness that melts at a rate of 50 cm /min. W en the thickness of ice 5 cm, then the rate at which thickness of ice decreases in cm/min 6 π ( b ) 8 π ( c ) 54 π d ) 5 6 π [ AIEEE 005 ] ( 9 ) Let f : R R be a differenti ble function having f ( ) = 6, f ( ) =. Then 48 f( ) 4t dt equa s ( b ) 6 ( c ) ( d ) 8 [ AIEEE 005 ] ( 0 ) If the e u tion a n n + a n - n a = 0, a 0, n has a ( ) If pos tive root = α, then the equation n a n n - + ( n - ) a n - n a = 0 has positive root which greater than α ( b ) smaller than α ( c ) greater than or equal to α ( d ) equal to α [ AIEEE 005 ] a b + + = e, then the values of a and b are a R, b R ( b ) a =, b R ( c ) a R, b = ( d ) a =, b = [ AIEEE 004 ]

3 08 - DIFFERENTIAL CALCULUS Page ( ) Let f ( ) = π then f 4 - tan, 4 - π ( b ) ( ) If = ( c ) - y + e y +... e, > 0, then + ( b ) ( c ) π π π, 4 0,. If f ( ) continuous in 0,, - ( d ) - [ AIEEE 004 ] dy d ( d ) + [ AIEEE 004 ] ( 4 ) A point on the parabola y = 8 at which the ordinate increases at twice the rate of the abscsa (, 4 ) ( b ) (, - 4 ) ( c ) -, ( d ), 8 8 [ AIEEE 004 ] ( 5 ) A function y = f ( ) has a second order derivative f ( ) = 6 ( - ). If its graph passes through the poin (, ) and at that point the tangent to the graph y = - 5, then the unction ( - ( b ) ( - ) ( c ) ( + ) ( d ) ( + ) [ AIEEE 004 ] ( 6 ) he normal to the curve = a ( + cos θ ), y = a sin θ at θ always passes through the fied point ( a, 0 ) ( b ) ( 0, a ) ( c ) ( 0, 0 ) ( d ) ( a, a ) [ AIEEE 004 ] ( 7 ) If a + b + 6c = 0, then at least one root of the equation a + b + c = 0 lies in the interval ( 0, ) ( b ) (, ) ( c ) (, ) ( d ) (, ) [ AIEEE 004 ]

4 08 - DIFFERENTIAL CALCULUS Page 4 ( 8 ) Let f ( ) be a polynomial function of second degree. If f ( ) = f ( - ) and a, b, c are in A. P., then f, f ( b ) and f ( c ) are in ( 9 ) ( 0 ) If A. P. ( b ) G. P. ( c ) H. P. ( d ) A. G. P. [ AIEEE 00 ] π - tan + tan [ - sin ] [ π - ] 0 ( b ) ( c ) log ( + ) - log ( - ) 0 0 ( b ) - ( ) If f( ) = ( c ) log ( + a ) - log ( - b ) = ( d ) 8 = k then the value f k ( d ) - [ AIEEE 00 ] [ AIEEE 00 ] continuous at = 0, then the value of f ( 0 ) ab ( b ) a + b ( c ) a - b ( d ) log a - log b [ AIEEE 00 ] ( ) If y = , then the value of!!! 0 ( b ) ( c ) ( d ) y [ AIEEE 00 ] ( ) The value of + n n... + n 4 dy d zero ( b ) 4 ( c ) 5 ( d ) 0 [ AIEEE 00 ]

5 08 - DIFFERENTIAL CALCULUS Page 5 ( 4 ) If f : R R satfies f ( + y ) = f ( ) + f ( y ), for all, y R and f ( ) = 7, n then the value of f ' ( r ) 7n r = ( b ) 7 n ( n + ) ( c ) 7 ( n + ) ( d ) 7 n ( n + ) [ AIEEE 00 ] ( 5 ) The real number when added to its inverse gives the min mum value of the sum at equal to ( b ) - ( c ) ( d ) - [ AIEEE 00 ] ( 6 ) If the function f ( ) = - 9a + a + where a > 0, attains its maimum and minimum at p and q respectively such that p = q, then a equals ( b ) ( c ) ( d ) 4 [ AIEEE 00 ] ( 7 ) If f ( ) = n, then the val e of f ( ) - f ' ( )! + f " ( )! + f ''' ( )! ( - n n ) f ( ) n! n ( b ) n ( c ) ( d ) 0 [ AIEEE 00 ] ( 8 ) If = t π + t and y = sin t π ( b ) - 6 π ( c ) π π + cos t, then at t =, the value of ( d ) - 4 π ( 9 ) If = cos θ - cos θ and y = sin θ - sin θ, then the value of dy d dy d [ AIEEE 00 ] sin θ ( b ) cos θ ( c ) tan θ ( d ) cot θ [ AIEEE 00 ]

6 08 - DIFFERENTIAL CALCULUS Page 6 ( 0 ) Let f = g = k and their nth derivatives f n, g n et and are not equal for some n. Further if f g ( ) - f - g f ( ) + g a g ( ) - f ( ) = 4, then the value of k 4 ( b ) ( c ) ( d ) 0 [ AIEEE 00 ] ( ) The value of 0 0 ( b ) ( - 0 cos ) sin 5 sin ( c ) sin α - sin β ( ) The value of α β α - β 0 ( b ) ( c ) ( ) The value of l m sin β β cos ( d ) ( d ) 5 6 sin β β [ AIEEE 00 ] [ AIEEE 00 ] 0 ( b ) ( c ) ( d ) does not et [ AIEEE 00 ] ( 4 ) f ( ) = on [ -, 4 ], then relative maimum occurs at = ( a - ( b ) - ( c ) ( d ) 4 [ AIEEE 00 ] ( 5 ) If f ( ) = - e 0, +, 0, then f ( ) = 0 dcontinuous everywhere ( b ) continuous as well as differentiable for all ( c ) neither differentiable nor continuous at = 0 ( d ) continuous at all but not differentiable at = 0 [ AIEEE 00 ]

7 08 - DIFFERENTIAL CALCULUS Page 7 ( 6 ) If y a twice differentiable function and cos y + y cos = π, then y ( 0 ) = π ( b ) - π ( c ) 0 ( d ) [ IIT 005 ] ( 7 ) f ( ) = l l l - l not differentiable at = 0, ± ( b ) ± ( c ) 0 ( d ) [ IIT 005 ] ( 8 ) If f a differentiable function such that f : R R f n then = 0 n I, n, f ( ) = 0 [ 0, ] ( b ) f ( 0 ) = 0, but f ( 0 ) may or may not be 0 ( c ) f ( 0 ) = 0 = f ( 0 ) ( d ) l f ( ) l [ 0, ] [ IIT 005 ] ( 9 ) f a twice differentiable polynomial function of such that f ( ) =, f ( ) = 4 and f ( ) = 9, then f ( ) =, R ( b ) f ( ) = f ( ) = 5, [, ] ( c ) f ( ) = for only [ ] ( d ) f ( ) =, (, ) [ IIT 005 ] [ Note: Th question should have been better put as polynomial function of degree two rather than wice differentiable function. ] ( 40 ) S a set of polynomial of degree less than or equal to, f ( 0 ) = 0, f ( ) =, f ( ) 0, [ 0, ], then set S = ( a a + ( - a ), a R ( b ) a + ( - a ), 0 < a < c ) a + ( - a ), 0 < a < ( d ) φ [ IIT 005 ] ( 4 ) Let y be a function of, such that log ( + y ) = y, then y ( 0 ) 0 ( b ) ( c ) ( d ) [ IIT 004 ] ( 4 ) Let f ( ) = α log for > and f ( 0 ) = 0 follows Rolle s theorem for [ 0, ], then α - ( b ) - ( c ) 0 ( d ) [ IIT 004 ]

8 08 - DIFFERENTIAL CALCULUS Page 8 ( 4 ) If f ( ) strictly increasing and differentiable, then f ( ) 0 f ( ) - - f ( ) f ( 0 ) ( b ) - ( c ) 0 ( d ) [ IIT 004 ] ( 44 ) Let f ( ) = + b + c + d, 0 < b < c, then f ( ) strictly increasing ( b ) has local maima ( c ) has local minima ( d ) a bounded curve [ IIT 004 ] ( 45 ) If f ( ) a differentiable function, f ( ) =, ( ) 6, where f ( c ) means the derivative of the function at = c, then f ( + h + h ) - h 0 f ( + h - h ) - f ( ) f ( ) does not et ( b ) - ( c ( d ) ( 46 ) If sin n [ ( a - n ) n 0 a equal to n + n ( b ) n ( c ) - tan ] n [ IIT 00 ] = 0, where n a non-zero positive integer, then ( d ) n + n ( 47 ) Which functi n does not obey Mean Value Theorem in [ 0, ]? [ IIT 00 ] -, < sin, 0 f ( ) = ( b ) f ( ) = -,, = 0 ( c ) f ( ) = l l ( d ) f ( ) = l l [ IIT 00 ] ( 48 ) The domain of the derivative of the function f ( ) = tan -, ( l l - ), if if l l l l > R - { 0 } ( b ) R - { } ( c ) R - { - } ( d ) R - { -, } [ IIT 00 ]

9 08 - DIFFERENTIAL CALCULUS Page 9 ( 49 ) The integer n for which 0 ( cos - ) ( cos - n e ) a finite non-zero number ( b ) ( c ) ( d ) 4 [ I T 00 ] ( 50 ) If f : R R be such that f ( ) = and f ( ) = 6, then f ( + ) 0 f ( ) equals ( b ) e ( c ) e ( d ) e [ IIT 00 ] ( 5 ) The point ( s ) on the curve y + = y wh re the tangent vertical, / ( are ) ± 4, - ( b ) ±, 0 ( c ) ( 0, 0 ) ( d ) ± 4, [ IIT 00 ] ( 5 ) Let f : R R be a functio defined by f ( ) = {, }. The set of all points where f ( ) not differentiable { -, } ( ) { -, 0 } ( c ) { 0, } ( d ) { -, 0, } [ IIT 00 ] ( 5 ) The left hand derivative of f ( ) = [ ] sin ( π ) at = k, where k an integer, ( - ) k ( k - ) π ( b ) ( - ) k - ( k - ) π k ( c ) ( - k π ( d ) ( - k ) - k π [ IIT 00 ] 54 The left hand derivative of f ( ) = [ ] sin ( π ) at = k, where k an integer, ( - ) k ( k - ) π ( b ) ( - ) k - ( k - ) π ( c ) ( - ) k k π ( d ) ( - ) k - k π [ IIT 00 ] ( 55 ) 0 sin ( π cos ) equals - π ( b ) π ( c ) π / ( d ) [ IIT 00 ]

10 08 - DIFFERENTIAL CALCULUS Page 0 ( 56 ) If f ( ) = e ( - ), then f ( ) increasing on -, ( b ) decreasing on R ( c ) increasing on R ( d ) decreasing on -, [ IIT 00 ] ( 57 ) Which of the following functions differentiable at = 0? cos ( l l ) + l l ( b ) cos ( l l ) - l l ( c ) sin ( l l ) + l l ( d ) sin ( l l ) - l l [ IIT 00 ] ( 58 ) If + y =, then yy - ( y ) + = 0 ( b ) yy + ( y ) + = 0 ( c ) yy + ( y ) - = 0 ( d ) yy + ( y ) + = 0 [ IIT 000 ] ( 59 ) For R, - = + e ( b ) e - ( c ) e - 5 ( d ) e 5 [ IIT 000 ] ( 60 ) Consider the fo owing statements in S and R: π S: Both sin and cos are decreasing functions in the interval, π R: If a differentiable function decreases in an interval ( a, b ), then its derivative also decreases in ( a, b ). Which of the following true? ( a Both S and R are wrong. ( b ) Both S and R are correct, but R not the correct eplanation of S. ( c ) S correct and R correct eplanation of S. ( d ) S correct and R wrong. [ IIT 000 ] ( 6 ) If the normal to the curve y = f ( ) at the point (, 4 ) makes an angle positive X-a, then f ( ) = π 4 with the - ( b ) - / 4 ( c ) 4 / ( d ) [ IIT 000 ]

11 ( 6 ) If f ( ) = l l 08 - DIFFERENTIAL CALCULUS Page for for 0 < l l = 0, then at = 0, f has a local maimum ( b ) no local maimum ( c ) a local minimum ( d ) no etremum [ I T 000 ] ( 6 ) For all ( 0, ), which of the following true? e < + ( b ) log e ( + ) < ( c ) sin > ( d ) log e > [ IIT 000 ] ( 64 ) The function f ( ) = sin 4 + cos 4 increases if 0 < < 8 π ( c ) π 5π < < 8 8 ( b ) ( d ) π π < < 4 8 5π π < < 8 4 [ IIT 999 ] ( 65 ) The function f ( ) = [ ] - [ ] where [ y ] the greatest integer less than or equal to y, dcontinuous at all integers ( b ) all integers ecept 0 and ( c ) all integers e p 0 ( d ) all integers ecept [ IIT 999 ] ( 66 ) The function f ( ) = ( - ) l - + l + cos ( l l ) NOT differentiable at ( 67 ) - ( b ) 0 ( c ) ( d ) [ IIT 999 ] tan 0 - tan ( - cos ) ( b ) - ( c ) = ( d ) - [ IIT 999 ] t 5 ( 68 ) The function f ( ) = t ( e - ) ( t - ) ( t - ) ( t - ) dt has a local minimum at = - 0 ( b ) ( c ) ( d ) [ IIT 999 ]

12 ( 69 ) - cos ( DIFFERENTIAL CALCULUS Page ) ets and equal to ( b ) ets and equal to - ( c ) does not et because - 0 ( d ) does not et because left hand it right hand it [ IIT 998 ] ( 70 ) If ( t ) dt = + 0 f t f ( t ) dt, then the value of f ( ) ( b ) 0 ( c ) ( d ) - ( 7 ) Let h ( ) = min [, ], for every real number, then [ IIT 998 ] h continuous for all ( b ) h differentiable for all ( c ) h ( ) = for all > ( d ) h not differentiable at two values of [ IIT 998 ] ( 7 ) If h ( ) = f ( ) - [ f ( ) ] for every real number, then h increasing whenev r f increasing ( b ) h increasing wh ever f decreasing ( c ) h decreasing whenever f decreasing ( d ) nothing ca be said in general [ IIT 998 ] ( 7 ) If f ( ) = sin and g ( ) =, where 0 <, then in th interval tan ( ) both f ( ) and g ( ) are increasing functions ( b ) both f ( ) and g ( ) are decreasing functions ( c ) f ( ) an increasing function ( d ) g ( ) an increasing function [ IIT 997 ] n r ( 74 ) equals n p n n + r + 5 ( b ) ( c ) - + ( d ) + [ IIT 997 ]

13 ( 75 ) If f ( ) = 6 p 08 - DIFFERENTIAL CALCULUS Page sin cos d - 0, ( p a constant ), then [ f ( ) ] d p p at = 0 p ( b ) p + p ( c ) p + p ( d ) independent of p [ IIT 997 ] - ( 76 ) The function f ( ) = [ ] cos π, where [. ] denote the greatest integer function, dcontinuous at all ( b ) all integer points ( c ) no ( d ) which not an integer [ IIT 995 ] ( 77 ) If f ( ) defined and continuous for all > 0 and satfy f y all, y and f ( e ) =, then = f ( ) - f ( y ) for f ( ) bounded ( b ) 0 as 0 ( c ) f ( ) as 0 ( d ) f ( ) = log [ IIT 995 ] ( 78 ) On the interval [ 0, ], he function 5 ( - ) 75 attains maimum value at the point 0 ( b ) 4 ( c ) ( d ) [ IIT 995 ] ( 79 ) Th func ion f ( ) = l p - q l + r l l, ( -, ) where p > 0, q > 0, r > 0, assumes its minimum value only at one point if p q ( b ) r q ( c ) r p ( d ) p =q = r [ IIT 995 ] ( 80 ) The function f ( ) = ln ( π ln ( e + + ) ) increasing on [ 0, ) ( b ) decreasing on [ 0, ) π π ( c ) increasing on 0, and decreasing on e, e π π ( d ) decreasing on 0, and increasing on e, e [ IIT 995 ]

14 08 - DIFFERENTIAL CALCULUS Page 4 ( 8 ) The function f ( ) = ma { ( - ), ( + ), }, ( -, ), continuous at all points ( b ) differentiable at all points ( c ) differentiable at all points ecept at = and = - ( d ) continuous at all points ecept at = and = - [ IIT 995 ] ( 8 ) Let [. ] denote the greatest integer function and f ( ) = [ tan ] Then, f ( ) does not et ( b ) f ( ) contin ous t = 0 0 ( c ) f ( ) not differentiable at = 0 ( d ) f ( 0 ) = [ IIT 99 ] ( 8 ) If f ( ) = ,, - <, hen f ( ) increasing on [ -, ] ( b ) f ( ) continuous on [ -, ] ( c ) f ( ) maimum at = ( d f ( ) does not et [ IIT 99 ] ( 84 ) The value of 0 ( - cos ) ( b ) - ( ) 0 ( d ) none of these [ IIT 99 ] ( 85 ) The follow ng functions are continuous on ( 0, π ). π tan ( b ) t sin dt 0 t π π ( ), 0 < ( d ) sin, 0 < 4 π π π sin, < π sin ( π + ), < < π [ IIT 99 ] ( 86 ) If f ( ) = f ( ) -, then, on the interval [ 0, π ], tan [ f ( ) ] and are both continuous ( b ) f ( ) are both dcontinuous ( c ) f - ( ) are both continuous ( d ) f - ( ) are both dcontinuous [ IIT 989 ]

15 08 - DIFFERENTIAL CALCULUS Page 5 ( 87 ) If y = P ( ), a polynomial of degree, then d d d y y d equals P ( ) + P ( ) ( b ) P ( ) P ( ) ( c ) P ( ) P ( ) ( d ) a constant [ IIT 988 ] ( 88 ) The function f ( ) = continuous at = ( b ) differentiable at = ( c ) continuous at = ( d ) differentiable at = [ IIT 988 ] ( 89 ) The set of all points where the function f ( ) = < + l l differentiable ( -, ) ( b ) ( 0, ) ( c ) (, 0 ) ( 0, ) ( d ) ( 0, ) ( e ) none of these [ IIT 987 ] ( 90 ) Let f and g be increasing nd decreasing functions respectively from ( 0, ) to ( 0, ). Let h ( ) = f [ g ( ) ] If h ( 0 ) = 0, h ( ) - h ( ) always zero ( b ) always negative ( c ) always positive ( d ) strictly increasing ( ) none of these [ IIT 987 ] ( 9 ) Let P ( ) = a 0 + a + a a n n be a polynomial in a real variable with 0 < a 0 < a < a < < a n. The function P ( ) has neit er a maimum nor a minimum ( b ) only one maimum ( c ) only one minimum ( d ) only one maimum and only one minimum ( e ) none of these [ IIT 986 ] ( 9 ) The function f ( ) = + l sin l continuous nowhere ( b ) continuous everywhere ( c ) differentiable ( d ) not differentiable at = 0 ( e ) not differentiable at infinite number of points [ IIT 986 ] ( 9 ) Let [ ] denote the greatest integer less than or equal to. If f ( ) = [ sin π ], then f ( ) continuous at = 0 ( b ) continuous in ( -, 0 ) ( c ) differentiable at = ( d ) differentiable in ( -, ) ( e ) none of these [ IIT 986 ]

16 08 - DIFFERENTIAL CALCULUS Page 6 sin [ ] ( 94 ) If f ( ) =, [ ] 0 [ ] = 0, [ ] = 0, where ] denotes the greatest integer less than or equal to, then f ( ) 0 equals ( b ) 0 ( c ) - ( d ) none of these [ IIT 985 ] ( 95 ) If f ( ) = ( - + ), then ( 96 ) f ( ) continuous but not differentiable at = 0 ( b ) f ( ) differentiable at = 0 ( c ) f ( ) not differentiable at = 0 ( d ) none of these [ IIT 985 ] n n - n - n - n 0 ( b ) - ( c ) ( 97 ) If + l y l = y, then y as a function of qual to ( d ) none of these [ IIT 984 ] defined for all real ( b ) continuous at = 0 ( c ) differentiable for al dy ( d ) such that = for < 0 d [ IIT 984 ] ( 98 ) If G ( ) = - 5, then 4 ( b ) 5 G ( ) - - G ( ) has the value ( c ) - 4 ( d ) none of these [ IIT 98 ] ( 99 ) If f =, f =, g = -, g =, then the value of g ( ) f - g f ( ) a - a ( b ) 5 ( c ) 5 ( d ) none of these [ IIT 98 ] ln ( + a ) - ln ( - b ) ( 00 ) The function f ( ) = not defined at = 0. The value which should be assigned to f at = 0, so that it continuous at = 0, a - b ( b ) a + b ( c ) ln a + ln b ( d ) none of these [ IIT 98 ]

17 08 - DIFFERENTIAL CALCULUS Page 7 ( 0 ) The normal to the curve = a ( cos θ + θ sin θ ), y = a ( sin θ - θ cos θ ) at any point θ such that it makes a constant angle with the X-a ( b ) it passes through the origin ( c ) it at a constant dtance from the origin ( d ) none of these [ IIT 98 ] ( 0 ) If y = a ln + b + has its etremum values at = - and = then a =, b = - ( b ) a =, b = - ( c ) a = -, b = ( d ) none of these [ IIT 98 ] ( 0 ) There ets a function f ( ) satfying f ( 0 ) =, f ( 0 ) = -, f ( ) > 0 for all and f ( ) > 0 for all ( b - < f ( ) < 0 for all ( c ) - f ( ) - for all d ) f ( ) < - for all [ IIT 98 ] ( 04 ) For a real number y, let [ y ] denote the greatest integer less than or equal to y. Then tan [ π ( - π ) ] the function f ( ) = + [ ] dcontinuous a some ( b ) continuous at all, but the derivative f ( ) does not et for some ( c ) f ( ) ets for all, but the derivative f ( ) does not et for some ( d ) f ( ets for all [ IIT 98 ] ( 05 ) If f ( ) = - sin, then f ( ) + cos 0 ( b ) ( c ) ( d ) none of these [ IIT 979 ]

18 08 - DIFFERENTIAL CALCULUS Page 8 Answers c d c a c a b b d b b c c d b a a a c c b d c d c c d b d a a d d d d a a c a b b d b a c d a d c c d d a a b a d b c d d d b b b d c b,d d a a,c,d a c c d b d b c b a,c b a,b,c,d d b,c b c a,b,c a a c b,d,e a,d d a b a,b,d d c b c b a d c

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436)

HEAT-3 APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA MAX-MARKS-(112(3)+20(5)=436) HEAT- APPLICATION OF DERIVATIVES BY ABHIJIT KUMAR JHA TIME-(HRS) Select the correct alternative : (Only one is correct) MAX-MARKS-(()+0(5)=6) Q. Suppose & are the point of maimum and the point of minimum