Find the inflection points of the function $y=\frac{2x^5}{5}-5x^3+6$. Determine the intervals of concavity as well.

Inflection points: $x=\left\lbrace-\frac{\sqrt{15}}{2},0,\frac{\sqrt{15}}{2}\right\rbrace$

Concave down: $(-\frac{\sqrt{15}}{2}, 0) \cup (\frac{\sqrt{15}}{2}, \infty)$

Concave up: $(-\infty, -\frac{\sqrt{15}}{2}) \cup (0, \frac{\sqrt{15}}{2})$

Inflection points: $x=\left\lbrace-\frac{\sqrt{15}}{2},0,\frac{\sqrt{15}}{2}\right\rbrace$

Concave down: $(-\infty, -\frac{\sqrt{15}}{2}) \cup (0, \frac{\sqrt{15}}{2})$

Concave up: $(-\frac{\sqrt{15}}{2}, 0) \cup (\frac{\sqrt{15}}{2}, \infty)$

Inflection points: $x=\{-\frac{\sqrt{15}}{2},\frac{\sqrt{15}}{2}\}$

Concave down: $(0,\infty )$

Concave up: $(-\infty ,0)$

Inflection points: $x=\{0,\frac{\sqrt{15}}{2}\}$

Concave down: $\left(0,\infty\right)$

Concave up: $(-\infty,0)$