# What are the points of inflection of f(x)=x^{2}e^{11 x} ?

Oct 2, 2016

$x = \pm \sqrt{\frac{2}{11}} - \frac{2}{11}$.

#### Explanation:

Sufficient condition for giving points of inflexion, if any#:r

f''=0 and f''' is not 0.

Here,

$f ' = 2 x {e}^{11 x} + 11 {x}^{2} {e}^{11 x} = {e}^{11 x} \left(11 {x}^{2} + 2 x\right)$

$f ' ' = {e}^{11 x} \left(121 {x}^{2} + 44 x + 2\right)$

$f ' ' ' = {e}^{11 x} \left(1331 {x}^{2} + 726 x + 66\right)$

f''=0 gives

$x = \pm \sqrt{\frac{2}{11}} - \frac{2}{11}$

and $f ' ' ' \left(\pm \sqrt{\frac{2}{11}} - \frac{2}{11}\right)$ are not 0.

So, the points of inflexion are $x = \pm \sqrt{\frac{2}{11}} - \frac{2}{11}$