# What are the values and types of the critical points, if any, of f(x)=e^(-2x^2)-xe^x?

Jun 16, 2018

${P}_{\max} \left(- 0.182115825969939296277 , 1.087614208523047937841\right)$

#### Explanation:

We get
$f ' \left(x\right) = - {e}^{- 2 {x}^{2}} \left({e}^{x + 2 {x}^{2}} + 4 x + {e}^{x + 2 {x}^{2}} x\right)$
By a numerical method we get
${x}_{1} \approx - 0.182115825969939296277$
and
$f ' ' \left(x\right) = - {e}^{- 2 {x}^{2}} \left(4 + 2 {e}^{x + 2 {x}^{2}} + {e}^{x + 2 {x}^{2}} x - 16 {x}^{2}\right)$
and

$f ' ' \left({x}_{1}\right) < 0$
so we get a maximum.