# What are the points of inflection, if any, of f(x) = 5x^3 + 30x^2 - 432x ?

May 22, 2018

$\left(- 2 , 944\right)$

#### Explanation:

points of inflection are where $f ' ' \left(x\right)$ changes signs

in this problem, $f ' \left(x\right) = 15 {x}^{2} + 60 x - 432$, $f ' ' \left(x\right) = 30 x + 60$

since $f ' ' \left(x\right)$ is linear, you can find where it changes signs by setting it equal to 0.

$f ' ' \left(x\right) = 30 x + 60 = 0$
$x = - 2$

if you plug in x-values near -2, you can see that $f ' ' \left(x\right)$ changes signs at x=-2:

$f ' ' \left(- 1.9\right) = 3 > 0$
$f ' ' \left(- 1.99\right) = 0.3 > 0$

$f ' ' \left(- 2.1\right) = - 3 < 0$
$f ' ' \left(- 2.01\right) = - 0.3 < 0$

to find point of inflection: $f \left(- 2\right) = 944$
the point of inflection is $\left(- 2 , 944\right)$

check with the graph of $f \left(x\right)$:
graph{5x^3+30x^2-432x [-14, 10, -10000, 10000]}

it seems to change concavity around $x = - 2$