# How do you find all local maximum and minimum points given y=x^3-9x^2+24x?

Nov 7, 2016

Let $f \left(x\right) = {x}^{3} - 9 {x}^{2} + 24 x$

Find where $f \left(x\right)$ has a slope of zero. This is the same as finding points where $f ' \left(x\right)$ is equal to zero.

Find $f ' \left(x\right)$ using the power rule:
$f ' \left(x\right) = 3 {x}^{2} - 18 x + 24$

Find where $f ' \left(x\right)$ is equal to zero by factoring:
$0 = 3 \left[{x}^{2} - 6 x + 8\right]$
$0 = 3 \left(x - 4\right) \left(x - 2\right)$

$x = 4 , x = 2$

Plug in values in between these numbers to test whether each is a local max or min

$x = 2$ is a local maxima, and $x = 4$ is a local minima