Use derivatives to sketch the graph of a function

Let

$f(x) = (x-1)^2 (x+2).$

Find all points such that $f'(x) = 0$ ;
Determine the intervals on which $f$ is monotonic by examining the sign of $f'$ ;
Determine the intervals on which $f'$ is monotonic by examining the sign of $f''$ ;
Sketch the graph of $f$ .

We take the derivative,
$\begin{align*} f'(x) &= 2(x-1)(x+2) + (x-1)^2 \\ &= (x-1)(2x + 4 + x - 1) \\ &= (x-1)(3x + 3) \\ &= 3(x^2 -1) \end{align*}$

Thus,

$f'(x) = 0 \quad \implies \quad 3(x^2-1) = 0 \quad \implies \quad x = \pm 1.$
$f$ is increasing for $|x| > 1$ , and decreasing for $|x| < 1$ .
Taking the second derivative,
$f'(x) = 3(x^2-1) \quad \implies \quad f''(x) = 6x.$

Thus, $f'$ is increasing if $x > 0$ , and decreasing if $x < 0$ .
We sketch the curve,