# How do you find all points of inflection f(x)=x^3-12x?

Jul 1, 2015

Analyze the sign of the second derivative.

#### Explanation:

A point of inflection (aka an inflection point) for a function is a point of the graph of the function at which the concavity changes.
We find concavity by looking at the sign of the second derivative. So an inflection point can also be described as a point of the graph of the function at which the sign of the second derivative changes.

$f \left(x\right) = {x}^{3} - 12 x$

$f ' \left(x\right) = 3 {x}^{2} - 12$

$f ' ' \left(x\right) = 6 x$

In general, a function may change signs at values of $x$ where the function is $0$ or the function is discontinuous.

In this case, the function $f ' '$ is a polynomial, so it is never discontinuous. And obviously, $f ' ' \left(x\right) = 0$ at $x = 0$

Furthermore, $f ' '$ is negative for $x < 0$ and positive for $x > 0$. So the sign of $f ' '$ does change at $x = 0$.

Recalling that an infletion point is a point on the graph, we realize that we need the $y$ value at $x = 0$

$f \left(x\right) = {x}^{3} - 12 x$, so $f \left(0\right) = 0$

There is one point of inflection. It is $\left(0 , 0\right)$.