# What are the points of inflection, if any, of f(x) = x^3-8x^2+13 ?

May 24, 2017

Point of inflection is $x = \frac{8}{3}$

#### Explanation:

The point of inflection occurs at a point on a curve at which the curve changes from being concave to convex , or vice versa.

This occurs at a point where $f ' ' \left(x\right)$ i.e. $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = 0$.

As $f \left(x\right) = {x}^{3} - 8 {x}^{2} + 13$

$\frac{\mathrm{df}}{\mathrm{dx}} = 3 {x}^{2} - 16 x$

and $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = 6 x - 16$

and point of inflection is given by $6 x - 16 = 0$ i.e. $x = \frac{8}{3}$

graph{x^3-8x^2+13 [-10, 10, -110.7, 49.3]}