4
$\begingroup$

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

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

$\endgroup$
6
$\begingroup$

Hint: It is three independent single-variable calculus problems. Do you know how to maximize $te^{-t^2}$?

$\endgroup$
  • $\begingroup$ Hm... need to use derivatives and calculate global maximum? I'll try $\endgroup$ – Steve Feb 26 '13 at 8:12
  • 1
    $\begingroup$ 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 $3$. $\endgroup$ – André Nicolas Feb 26 '13 at 9:01
  • $\begingroup$ Thanks. I've solved this task :-) $\endgroup$ – Steve Feb 26 '13 at 9:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.