Finding f intervals How would I find the f intervals for the following two functions.
$f(x)=(x-2)^2(x+1)^2$
using the chain rule I got $(x-2)^2(2)(x+1)(1)+(2)(x-2)(1)(x+1)^2$
then I got f decrease $(-\infty,-1]$
and f increase $[2,\infty)$
but the area between -1 and 2 in confusing me.
My second function is 
$f(x)=x+\frac{4}{x^2}$
differentiating I got $\frac{x^3-8}{x^3}$
so I got f increase $(\infty,0)$ and $[2,\infty)$
f decrease $(0,2]$
but did I do this correctly.

As you did $f'(x)$ correctly, we have $$f'(x)=2(x-2)(x+1)(2x-1)$$  When:


*

*$x\le -1~~$,  $(x-2)\leq0$ and then two terms $(x+1)$ and $(2x-1)$ are negative. So $f'(x)$ is negative.

*$x\le\frac{1}2~~$, $(x-2)\leq0$ and $(x+1)>0$ and $(2x-1)<0$ is negative, so $f'(x)$ is positive.

*$x\le2~~$, $(x-2)\leq0$ and then two terms $(x+1)$ and $(2x-1)$ are positive, so $f'(x)$ is nagative again.

*And if $x>2$ all terms are positive and so $f'(x)$ is positive again.
