# Prove, that $xe^{-x^{2}}+ye^{-y^{2}}+ze^{-z^{2}}\leq\sqrt{\frac{9}{2e}}$

Please help me prove, that for all real $x,y,z$

$$xe^{-x^{2}}+ye^{-y^{2}}+ze^{-z^{2}}\leq\sqrt{\frac{9}{2e}}$$

Hint: It is three independent single-variable calculus problems. Do you know how to maximize $te^{-t^2}$?
• Derivative of $te^{-t^2}$ is $-2t^2e^{-t^2}+e^{-t^2}$. Set equal to $0$, solve. We get $t=1/\sqrt{2}$. So max is $\frac{1}{\sqrt{2}}e^{-1/2}$. Multiply by $3$. – André Nicolas Feb 26 '13 at 9:01