Question

Determine the intervals for which f(x) is increasing and decreasing. Identify the extreme points.
f(x)=(x^2-2x-8)^2

Answer 1

Ok dokey! First you'd want to solve for the derivative of F(x), which ends up being: 4 (x-1) (x^2-2 x-8)

Secondly you'd want to identify where this second derivative is positive and negative, which you can identify by finding the extrema, which are the roots of F'(x) (Where it is zero).
These roots end up being: -2, 1, and 4.
From -infinity to -2, it is decreasing, from -2 to 1 it is increasing, from 1 to 4 it is decreasing, and from 4 to infinity it is increasing.

The extreme are those points:
-2 is a relative minimum
1 is a relative maximum
4 is a relative minimum.

Answer 2

There are two inflection points.
Refer to the Mathematica attachment.