Answer
Critical point $x=1$, no endpoints.
At $x=1$, $y=2$ minimum.
Work Step by Step
Step 1. Given the function $y=\begin{cases} 4-2x\hspace1cm x\leq1 \\ x+1\hspace1.2cm x\gt1 \end{cases}$
We can find its derivative as
$y'=\begin{cases} -2\hspace1cm x\leq1 \\ 1\hspace1.4cm x\gt1 \end{cases}$
Step 2. The critical points can be found when $y'=0$ or undefined which happens when $x=1$. The domain of the function is $(-\infty,\infty)$ and there are no endpoints.
Step 3. At $x=1$, $y=2$ and we can identify this as a minimum (use local test points as necessary) as shown in the figure.
