# What are the points of inflection of f(x)=x^3 - 9x^2 + 8x ?

Jan 12, 2016

at $x = 3$

#### Explanation:

Points of inflection occur when the second derivative of a function shifts sign (goes from positive to negative). This can also be viewed as a shift in concavity.

Find the second derivative:

$f \left(x\right) = {x}^{3} - 9 {x}^{2} + 8 x$
$f ' \left(x\right) = 3 {x}^{2} - 18 x + 8$
$f ' ' \left(x\right) = 6 x - 18$

The second derivative's sign could shift when it equals zero.

$f ' ' \left(x\right) = 0$
$6 x - 18 = 0$
$x = 3$

There is a possible point of inflection when $x = 3$.

We can check by analyzing the sign around the point $x = 3$.

$\left(- \infty , 3\right)$

$f ' ' \left(0\right) = - 18 \leftarrow \text{negative}$

$3$

$f ' ' \left(3\right) = 0 \leftarrow \text{zero}$

$\left(3 , + \infty\right)$

$f ' ' \left(4\right) = 6 \leftarrow \text{positive}$

As you can see, the concavity shifts from negative (concave down) to positive (concave up) when $x = 3$. Thus, it is a point of inflection.

$f \left(x\right)$ graphed:

graph{x^3-9x^2+8x [-3, 10, -78.3, 27.1]}