Answer
$y(0)=0$ is a local and absolute minimum, no maximum.
There is no inflection point.
See graph.
Work Step by Step
Step 1. Given the function $y=x^{2/5}$, we have $y'=\frac{2}{5}x^{-3/5}$ and $y''=-\frac{6}{25}x^{-8/5}$
Step 2. The extrema happen when $y'=0$, undefined, or at endpoints. As $y'\ne0$ and there are no endpoints, the only point need to be considered is $x=0$.
Step 3. Check sign changes of $y'$ across the critical point $(0,0)$, $..(-)..(0)..(+)..$. We can identify $y(0)=0$ as a local and absolute minimum. The function has no maximum.
Step 4. The inflection points can be found when $y''=0$ or it does not exist. As $y''\ne0$, the only possible inflection point is $(0,0)$
Step 5. Examine the sign of $y''$ on different intervals. We have $..(-)..(0)..(-)..$, The function is concave down on both sides of $x=0$, thus it is not an inflection point.
Step 6. See graph.
