Question

Consider the following. f(x) = 4x3 − 6x2 − 24x + 4

(a) Find the intervals on which f is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing

(b) Find the local maximum and minimum values of f. (If an answer does not exist, enter DNE.) local minimum value local maximum value

(c) Find the intervals of concavity and the inflection points. (Enter your answers using interval notation.)

concave up concave down inflection point (x, y) =

Answer 1

f(x) = 4x^3 - 6x^2 - 24x + 4

f'(x) = 12x^2 - 12x - 24 = 12(x - 2)(x + 1)

f''(x) = 24x - 12 = 12(2x - 1)

(a) Increasing => f'(x) > 0 , interval = (-infinity , -1)U(2 , infinity)

Decreasing => f'(x) < 0 , interval = (-1 , 2)

(b) f(-1) = 18 , f(2) = -36

Local maximum value = 18

Local minimum value = -36

(c) Concave up => f''(x) > 0 , interval = (1/2 , infinity)

Concave down => f''(x) < 0 , interval = (-infinity , 1/2)

Inflection point => f''(x) = 0

point (x,y) = (1/2 , -9)