# What are the points of inflection, if any, of f(x) = x^5/20 - 5x^3 + 5 ?

Apr 20, 2018

The points of inflection are $x = - \sqrt{30} , 0 , \mathmr{and} \sqrt{30}$.

#### Explanation:

$f \left(x\right) = \left({x}^{5} / 20\right) - 5 {x}^{3} + 5$

Points of inflection are where the second derivative of a function is equal to 0, and where the second derivative of a function switches signs. So first you find the second derivative and set it equal to 0.

$f ' \left(x\right) = \left(5 {x}^{4} / 20\right) - 15 {x}^{2} = \left({x}^{4} / 4\right) - 15 {x}^{2}$
$f ' ' \left(x\right) = \left(4 {x}^{3} / 4\right) - 30 x = {x}^{3} - 30 x$
${x}^{3} - 30 x = 0$

Factor out the x.
$x \left({x}^{2} - 30\right) = 0$
so $x = 0$ and $x = \pm \sqrt{30}$

To check if these are all inflection points, you check if the signs change around them. I plugged in $x = - 6 , - 1 , 1$, and $6$, but you can do this with any points outside and between the three answers. The signs changed around all 3 points, so all 3 are points of inflection.