Answer
The inflection points are at $x=0, \pm\sqrt 3$.
Concave up on $x\lt -\sqrt 3$ and $0\lt x\lt \sqrt 3$.
Concave down on $x\gt \sqrt 3$.
Work Step by Step
We have
$$y=10x^3-x^5, \quad y'=30x^2-5x^4, \quad y''=60x-20x^3$$
The inflection points occur when $y''=60x-20x^3=20x(3-x^2)=0$, that is $x=0, \pm\sqrt 3$.
Concave up when $y''\gt 0$, which occurs at $x\lt -\sqrt 3$ and $0\lt x\lt \sqrt 3$.
Concave down when $y''\lt 0$, which occurs at $x\gt \sqrt 3$.
