Answer
See graph and explanations.
Work Step by Step
Step 1. See graph for the function $y=x^{1/3}+(x-1)^{1/3}$.
Step 2. We have $y'=\frac{1}{3}x^{-2/3}+\frac{1}{3}(x-1)^{-2/3}$
Step 3. It is not difficult to see that $y'=0$ does not have a solution. Thus the critical points are $x=0,1$ where $y'$ is undefined.
Step 4. Checking the signs of $y'$ across the critical points, we have $..(+)..(0)..(+)..(1)..(+)..$; thus the function increases over all its open intervals without extrema.
Step 5. As $x\to0^-, y'\to\infty$, $x\to0^+, y'\to\infty$, the function has a vertical tangent line at $x=0$ where $y'(0)$ is undefined.
Step 6. As $x\to1^-, y'\to\infty$, $x\to1^+, y'\to\infty$, the function has a vertical tangent line at $x=1$ where $y'(1)$ is undefined.
