# How do you find the inflection points of the graph of the function: f(x) = ((4 x)/e^(9 x))?

Jul 29, 2015

Inflection points are points on the graph at which the concavity changes. So investigate concavity.

#### Explanation:

To investigate concavity, we will look at the sign of the second derivative.

$f \left(x\right) = \frac{4 x}{e} ^ \left(9 x\right)$

$f ' \left(x\right) = \frac{4 {e}^{9 x} - 4 x {e}^{9 x} \cdot 9}{e} ^ \left(18 x\right)$

$= \frac{4 - 36 x}{e} ^ \left(9 x\right)$

$f ' ' \left(x\right) = \frac{\left(- 36\right) {e}^{9 x} - \left(4 - 36 x\right) {e}^{9 x} \cdot 9}{e} ^ \left(18 x\right)$

$= \frac{36 \left(9 x - 2\right)}{e} ^ \left(9 x\right)$

The factors $36$ and ${e}^{9 x}$ are always positive, so the sign of $f ' '$ is the same as the sign of $9 x - 2$ which changes at $x = \frac{2}{9}$

The point $\left(\frac{2}{9} , f \left(\frac{2}{9}\right)\right)$ is the only inflection point.

Since $f \left(\frac{2}{9}\right) = \frac{8}{9 {e}^{2}}$,

the inflection point is: $\left(\frac{2}{9} , \frac{8}{9 {e}^{2}}\right)$