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)$