Math Challenges

Solution:
\( \displaylines{\frac{df}{\differentialD x}=\left(1-2x^2\right)ye^{-x^2-y^2}\\ \frac{df}{\differentialD y}=\left(1-2y^2\right)xe^{-x^2-y^2}\\ \text{These equal 0 simultaneously at five points: }\left(0,0\right)\text{ and }\left(\pm\frac{1}{\sqrt2},\pm\frac{1}{\sqrt2}\right)} \)