Let $f:\Bbb{R} \to \Bbb{R}$ be a function defined by $f(x)=\frac{x^4}{4}+\frac{x^3}{3}-x^2$. Find the intervals on which $f$ is increasing and decreasing, and identify the local extrema of $f$.
My attempt:
We have that $f’(x)=x^3+x^2-2x$
If $x\in (-\infty,-2)$, then $f’(x)<0$.
If $x\in (-2,0)$, then $f’(x)>0$.
If $x\in (0,1)$, then $f’(x)<0$.
If $x\in (1,\infty)$, then $f’(x)>0$.
Hence, the intervals on which $f$ is increasing are $(-2,0)$ and $(1,\infty)$.
The intervals on which $f$ is decreasing are $(-\infty,2)$ and $(0,1)$.
We have that, $f(0)$ is a local maximum, because $f’(x)$ changes signs from $+$ to $-$.
$f(1)$ and $f(-2)$ are a local minimum, because $f’(x)$ changes signs from $-$ to $+$.
Is that true, please? Thanks.