# How do you find the x coordinates of all points of inflection, final all discontinuities, and find the open intervals of concavity for f(x)=x^4-8x^3?

Nov 15, 2016

points of inflection at $x = 0$ and $x = 4$
concave up for $\left(- \infty , 0\right)$ and $\left(4 , \infty\right)$
concave down for $\left(0 , 4\right)$

#### Explanation:

You need the second derivative for concavity/inflection points

$f ' \left(x\right) = 4 {x}^{3} - 24 {x}^{2}$

${f}^{' '} \left(x\right) = 12 {x}^{2} - 48 x$

$= 12 x \left(x - 4\right)$

${f}^{' '} \left(x\right) = 0$ at $x = 0$ and $x = 4$

Make a sign chart. To the left of $x = 0$, the second derivative is positive. To the right of $x = 4$, the second derivative is positive. Between $0$ and $4$ the second derivative is negative.