# What are the values and types of the critical points, if any, of f(x) =x^3 + 3x^2-24x?

$\left(2 , - 28\right)$ Minimum Point,
$\left(- 4 , 80\right)$ Maximum Point and $\left(- 1 , 26\right)$ Point of inflection

#### Explanation:

$f \left(x\right) = {x}^{3} + 3 {x}^{2} - 24 x$ the given
$f ' \left(x\right) = 3 {x}^{2} + 6 x - 24$ the first derivative

Let $f ' \left(x\right) = 0$
$3 {x}^{2} + 6 x - 24 = 0$
solving for x:
$x = 2$ and $x = - 4$
when $x = 2$ , $y = - 28$ Minimum point
when $x = - 4$, $y = 80$ Maximum Point

Solving for $f ' ' \left(x\right)$
$f ' ' \left(x\right) = 6 x + 6$
set $f ' ' \left(x\right) = 0$ and solving for x:
$6 x + 6 = 0$
$x = - 1$
when $x = - 1$, $y = 26$ Point of inflection