Consider the equation below. f(x) = 4x3 + 12x2 - 96x + 8 (a) Find the intervals
ID: 2897917 • Letter: C
Question
Consider the equation below.f(x) = 4x3 + 12x2 - 96x + 8
(a) Find the intervals on which f is increasing. (Enter your answer using interval notation.)
Find the interval on which f is decreasing. (Enter your answer using interval notation.)
(b) Find the local minimum and maximum values of f.
local minimum value
local maximum value
(c) Find the inflection point.
(x, y) =
Find the interval on which f is concave up. (Enter your answer using interval notation.)
Find the interval on which f is concave down. (Enter your answer using interval notation.)
Explanation / Answer
f is increasing if f'(x) >0
we put
f'(x) = 12x2 + 24x -96 = 0
(x+4)(x-2) = 0
hence f'(x) is positive in (-4,2) hence f(x) is increasing for (-4,2)
f'(x) is negative for (-inf , -4) U (2, inf) , hence function is decreasing in this range
Max value of f when f'(x) = 0
at -4 and 2
at x = -4 we get f(x) = 524 (maximum value of f)
at x = 2 we get f(x) = -104 (minimum value of f)
Inflection point when f''(x) = 0
f''(x) = 24x + 24 = 0
x = -1 = point of inflection
Concave down => f(x) decreasing in (-inf , -4) U (2, inf)
concave up => f(x) increasing in (-4,2)