Answer
- no extrema.
- reflection point is $(0,0)$,
- concave up on $(-\infty,0)$ and concave down on $(0,\infty)$.
See graph.
Work Step by Step
Step 1. Given the function $y=x^{1/5}$, we have $y'=\frac{1}{5}x^{-4/5}$ and $y''=-\frac{4}{25}x^{-9/5}$
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 $x=0$. However, this is not an extreme because $y'\lt0$ and the function will increase when $x$ increases. Thus the function has no extrema.
Step 3. The inflection points can be found when $y''=0$ or it does not exist. As $y''\ne0$, the only reflection point is $(0,0)$
Step 4. To identify the intervals on which the functions are concave up and concave down, we need to examine the sign of $y''$ on different intervals. We have $..(+)..(0)..(-)..$,
Step 5. The function is concave up on $(-\infty,0)$ and concave down on $(0,\infty)$.
Step 6. See graph.
