# How do you find the local extrema for f(x)=-0.12x^3 + 900x - 830?

Feb 22, 2017

local max. at $\left(50 , 29170\right)$
local min. at $\left(- 50 , - 30830\right)$

#### Explanation:

$f \left(x\right) = - .12 {x}^{3} + 900 x - 830$

Find the first derivative: $f ' \left(x\right) = - .36 {x}^{2} + 900$

Find the critical numbers ($f ' \left(x\right) = 0$):

.36x^2 = 900; x^2 = 2500; x = +-50

Second derivative test:

f''(x) = -.72x;

$f ' ' \left(- 50\right) > 0$; relative min at $x = - 50$ and $f \left(- 50\right) = - 30 , 830$

$f ' ' \left(50\right) < 0$; relative max at $x = 50$ and $f \left(50\right) = 29 , 170$

local max. at $\left(50 , 29170\right)$
local min. at $\left(- 50 , - 30830\right)$