Find the relative maxima and relative minima, if any, of the function.
A)No relative minima; Relative maximum f (3) = - 5
B)Relative maximum: f (3) = -22; Relative minimum f (0) = - 5
C)Relative minimum: f (3) = -22; No relative maxima
D)No relative maxima or minima

Question

Find the relative maxima and relative minima, if any, of the function. 
A)No relative minima; Relative maximum f (3) = - 5 
B)Relative maximum: f (3) = -22; Relative minimum f (0) = - 5 
C)Relative minimum: f (3) = -22; No relative maxima 
D)No relative maxima or minima

Quizplus · Accepted Answer

To find relative maxima and minima, we need to find the critical points. Taking the derivative of the given function, we get: f'(x) = 2x^3 - 9x^2 + 12x Setting f'(x) = 0, we get: 2x^3 - 9x^2 + 12x = 0 2x(x-3)(x-2) = 0 The critical points are x = 0, x = 2, and x = 3. To determine whether these points are relative maxima or minima, we need to examine the sign of f'(x) near each point. For x < 0, f'(x) < 0 For 0 < x < 2, f'(x) > 0 For 2 < x < 3, f'(x) < 0 For x > 3, f'(x) > 0 Therefore, f(3) is a relative minimum, and there are no other relative extrema. The correct answer is C.

Find the Relative Maxima and Relative Minima, If Any, of the Function