# How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for y=-x^5+2x^3+4?

Jun 24, 2018

No discontinuities.
Points of inflection at $x = 0 , \pm \sqrt{\frac{3}{5}}$
CCU on (inf,$- \sqrt{\frac{3}{5}}$) and (0,$\sqrt{\frac{3}{5}}$)
CCD on ($- \sqrt{\frac{3}{5}}$,0) and ($\sqrt{\frac{3}{5}}$,inf)

#### Explanation:

$f \left(x\right)$ is a polynomial so continuous on reals.

By power rule we can find second derivative to be

$f ' ' \left(x\right) = - 20 {x}^{3} + 12 x$

Setting equal to zero we can find Possible Inflection Points

$- 20 {x}^{3} + 12 x = 0$
$x = 0 , \pm \sqrt{\frac{3}{5}}$

Testing concavity between points using $x = - 1 , - \frac{1}{2} , \frac{1}{2} , 1$ gives positive concavity between -inf and $- \sqrt{\frac{3}{5}}$ negative concavity from $- \sqrt{\frac{3}{5}}$ to 0 positive concavity from 0 to $\sqrt{\frac{3}{5}}$ and negative concavity from $\sqrt{\frac{3}{5}}$ to inf.