How do you find all points of inflection given y=-x^5+2x^3+4?

Dec 21, 2017

Inflection points at $x = 0$ and $x = \pm \sqrt{\frac{5}{3}}$

Explanation:

Find the second derivative:

$y ' = - 5 {x}^{4} + 6 {x}^{2}$

$y ' ' = - 20 {x}^{3} + 12 x$

This will have points of inflection when $y ' ' = 0$.

$0 = - 20 {x}^{3} + 12 x$

$0 = - 4 x \left(5 {x}^{2} - 3\right)$

This means that $x = 0$ or $x = \pm \sqrt{\frac{5}{3}}$.

Hopefully this helps!