Answer
$ a.\quad$ critical points at x=0 and x=1
$b.\quad $increasing on $(-\infty, 0)$ and $(1, \infty)$ , decreasing on $(0,1)$.
$c.\quad $local maximum at $x=0$ and a local minimum at $x=1$
Work Step by Step
$(a)$
$f'$ is defined everywhere.
$f'(x)=0$ for $x=0,1\qquad $... critical points at x=0 and x=1.
$(b)$
$\left[\begin{array}{cccccc}
& -\infty & & 0 & & 1 & & \infty\\\\
\text{test point} & & -1 & | & 0.5 & | & 2 & \\
\text{evaluate }f' & & +2 & | & -0.25 & | & +2 & \\
\text{sign of }f' & & + & | & - & | & + & \\
\text{behavior of }f(x) & & \nearrow & max & \searrow & min & \nearrow &
\end{array}\right]$
f is increasing on $(-\infty, 0)$ and $(1, \infty)$ , decreasing on $(0,1)$.
$(c)$
From the table, we see that f has:
- a local maximum at $x=0$ and
- a local minimum at $x=1$
