Answer
See graph and explanations.
Work Step by Step
Step 1. See graph for the function $y=x^{2/3}+(x-1)^{2/3}$.
Step 2. We have $y'=\frac{2}{3}x^{-1/3}+\frac{2}{3}(x-1)^{-1/3}$
Step 3.Letting $y'=0$, we can get $x=\frac{1}{2}$. Thus the critical points are $x=0,\frac{1}{2},1$, including points where $y'$ is undefined.
Step 4. Checking the signs of $y'$ across the critical points, we have $..(-)..(0)..(+)..(\frac{1}{2})..(-)..(1)..(+)..$; thus the function decreases over $(-\infty,0),(\frac{1}{2},1)$ and increases over $(0,\frac{1}{2}), (1,\infty)$. Thus $y(\frac{1}{2})=\sqrt[3] 2$ is a local maximum and $y(0)=y(1)=1$ are local maxima (also as absolute).
Step 5. As $x\to0^-, y'\to-\infty$, $x\to0^+, y'\to\infty$, the function has a cusp at $x=0$ where $y'(0)$ is undefined.
Step 6. As $x\to1^-, y'\to-\infty$, $x\to1^+, y'\to\infty$, the function has another cusp at $x=1$ where $y'(1)$ is undefined.
