# What are the extrema of f(x) = x^3 - 27x?

Nov 1, 2015

$\left(- 3 , 54\right) \mathmr{and} \left(3 , - 54\right)$

#### Explanation:

Relative maximum and minimum points occur when the derivative is zero, that is when $f ' \left(x\right) = 0$.
So in this case, when $3 {x}^{2} - 27 = 0$
#=>x=+-3.

Since the second derivative $f ' ' \left(- 3\right) < 0 \mathmr{and} f ' ' \left(3\right) > 0$, it implies that a relative maximum occurs at x=-3 and a relative minimum at x=3.

But: $f \left(- 3\right) = 54 \mathmr{and} f \left(3\right) = - 54$ which implies that $\left(- 3 , 54\right)$ is a relative maximum and $\left(3 , - 54\right)$ is a relative minimum.

graph{x^3-27x [-115.9, 121.4, -58.1, 60.5]}