Calculus Question

Question

Find the local maximum and minimum values of f using both the First and Second Derivative Tests.

f(x) = 7 + 6x² − 4x³

local maximum value
local minimum value

Answer 1

First we need to find the critical points using the first derivative:

f'(x) = 12x - 12x² = 12x ( 1 - x )

Setting f'(x) = 0, we find that the critical points are 0 and 1.

Next we need to plug these critical points into the second derivative to see if they are maximums or minimums.

f''(x) = 12 - 24x

f''(0) = 12

f''(1) = -12

So based on the second derivative test, there is a local minimum at x = 0 and a local maximum at x = 1.

Answer 2