Calculus Maimus Increasing, Decreasing, and st Derivative Test Show all work. No calculator unless otherwise stated. Multiple Choice = /5 + _ /5 over. Determine the increasing and decreasing open intervals of the function f ( ) ( 3) ( ) its domain. (A) Inc:,, Dec:, (C) Inc: (, ) ( 3, ), Dec: (, 3), Dec:,3, 3,, Dec:,3, 3, (B) Inc:, ( 3, ) (D) Inc: ( ),3 3, 5 (E) Inc: ( ), Dec: ( ). Let f be the function defined by ( ) cos, f = π π. Determine all open interval(s) on which f is decreasing. 5 7 (A) π, π π 5 (B) π, π π 5 (C) π 3, π π 6 6 8 8 (D) π, π π (E) 5π 7π π,,, π 6 6 = + _ 5. (i) Determine the derivative, 3. Let f ( ) f ʹ ( ). (A) ( (B) ( (C) ( (D) ( (E) ( f ) ( f ) ( = + )( 5 ) = + )( ) 5+ ) + ) ) f ) =( )( f ) =( )( f ) =( )( (ii) Find the open interval(s) on which f is increasing. (A) (, ) (, ) (B) (, 5) ( 5, ) (C) (, ) (D) ( (E) (,) (F) ( 5, 5), ) (, )
. The derivative of a function f is given for all by f ( ) ( 6) ( g ( ) ) ʹ = + + where g is some unspecified function. At which value(s) of will f have a local maimum? (A) = (B) = (C) = (D) = (E) =, 5. Which of the following statements about the absolute maimum and absolute minimum values of 6 f ( ) = 3 on the interval [0, ) are correct? (Hint: What type of discontinuity does + f ( ) have???) Free Response 9 (A) Ma = 3, No Min (B) No Ma, Min = (C) Ma = 3, Min = 9 (D) Ma = 5, No Min (E) No Ma, Min = 6. For each of the following, find the critical values (on the indicated intervals, if indicated.) a) f ( ) = (3 ) b) f ( ) = + c) f ( ) = d) () 3 f = 9 e) f ( ) = cos (), [0, π ] f) f ( ) = sin + sin, [0, π ] (g) f ( ) 7. Assume that f is differentiable for all. The signs of f ʹ are as follows. ) > 0 on (, ) (6, )and ) < 0 on (,6) Supply the appropriate inequality for the indicated value of c in the bo. Function Sign of gʹ ( c) a) g( ) = f ( ) + 5 b) g( ) = 3f ( ) 3 gʹ ( 5) 0 c) g( ) = f () gʹ ( 6) 0 d) g( ) = f () e) g( ) = f ( 0) f) g( ) = f ( 0) gʹ (8) 0 sin =, [0, π ] + cos
Calculus Maimus Concavity and the Second Derivative Test 3. If a < 0, the graph of y = a + 3 + + 5 is concave up on (A), _ (B), _ (C) _, (D) _, a a a a (E) (, ). If f ( ) ) fʹ ʹ ( ) 0 = 0 = 0 = 0, which of the following must be true about the graph of f? (A) There is a local ma at the origin (B) There is a local min at the origin (C) There is no local etremum at the origin (D) There is a point of inflection at the origin (E) There is a horizontal tangent at the origin 5 3. The -coordinates of the points of inflection of the graph of y = 5 + 3+ 7 are (A) 0 only (B) only (C) 3 only (D) 0 and 3 (E) 0 and. Which of the following conditions would enable you to conclude that the graph of f has a point of inflection at = c? (A) There is a local ma of f ʹ at = c (B) fʹ ʹ ( c) = 0 (C) fʹ ʹ ( c) does not eist (D) The sign of f ʹ changes at = c (E) f is a cubic polynomial and c = 0 5. Let f be a twice-differentiable function on (, ) such that the equation f ( ) = 0 has eactly 3 real roots, all distinct. Consider the following possibilities: I. the equation ) = 0 has at most 3 distinct roots. II. the equation fʹ ʹ ( ) = 0 has at least root. Which of these properties will f have? (A) I only (B) II only (C) I and II (D) neither I nor II 6. If f is a continuous function on ( 5,3) whose graph is given at right, which of the following properties are satisfied? I. fʹ ʹ ( ) > 0 on (,) II. f has eactly local etrema III. f has eactly critical points (A) I, II, and III (B) II only (C) I and III (D) I and II (E) III only (F) II and III (G) I only
Calculus Maimus Free Response 7. The figure above shows the graph of f ʹ, the derivative of the function f, for 7 7. The graph of f ʹ has horizontal tangent lines at = 3, =, and = 5, and a vertical tangent line at = 3. (a) Find all the values of, for 7< < 7, at which f attains a relative minimum. Justify your answer. (b) Find all the values of, for 7< < 7, at which f attains a relative maimum. Justify your answer. (c) Find all the values of, for 7 7 ʹ ʹ <. < <, at which f ( ) 0 (d) At what value of, for 7< < 7, does f attain its absolute maimum? Justify your answer. 8. Let h be a function defined for all 0 such that h( ) = 3 and the derivative of h is given by ) hʹ ( =, 0. (a) Find all values of for which the graph of h has a horizontal tangent, and determine whether h has a local maimum, a local minimum, or neither at each of these values. Justify your answers. (b) On what intervals, if any, is the graph of h concave up? Justify your answer. (c) Write an equation for the tangent to the graph of h at =. (d) Does the line tangent to the graph of h at = lie above or below the graph of h for >? Why? 3 9. A cubic polynomial function f is defined by f ( ) = +a +b+k, where a, b, and k are constants. The function f has a local minimum at =, and the graph of f has a point of inflection at =. Find the values of a and b. 0. For each of the following, i) find the open intervals on which f is increasing or decreasing, ii) find the local ma and min values of f, and iii) find the intervals of concavity and inflection points. (a) f ( ) = (b) f ( ) = cos sin, 0 π (c) f ( ) = ln + 3
. Find the local ma and min values of f ( ) = + using both the First and Second Derivative Tests. Which method do you prefer?. Sketch the graph of a function that satisfies all of the following conditions. (a) (b) ) > 0 for all, VA at =, fʹ ʹ ( ) > 0 if < or > 3, and fʹ ʹ ( ) < 0 if ) > 0 if <, ) < 0 if >, f ʹ ( ) 0 0< < 3, and fʹ ʹ ( ) > 0 if > 3 =, lim f( ) =, f ( < < 3. ) = f ( ), fʹ ʹ ( ) < 0 if