Question

Find all critical numbers for the following function. Then use the​ second-derivative test on each critical number to determine whether it leads to a local maximum or minimum.

Answer 1

Critical values are values where f'(x)=0 and the bounds of a function. Thus, let's solve for f'(x)!

f(x)=2x^3-3x^2+3x+8
f'(x)=6x^2-6x+3

Now let's set f'(x)=0

0=6x^2-6x+3
0=2x^2-2x+1

As it turns out, 2x^2-2x+1 isn't factorable!
This saves me some time because this means there are no critical numbers!