In Search of a More Elegant Solution I was asked to determine the maximum and minimum value of $$f(x,y,z)=(3x+4y+5z^{2})e^{-x^{2}-y^{2}-z^{2}}$$ on $\mathbb{R}^{3}$. 
Now, I employed the usually strategy; in other words calculating the partial derivatives, setting each to zero, and the solve for $x,y,z$ before comparing the values of the stationary points. I obtained $$M=5e^{-3/4}$$ as the maximum value and $$m=(-5e^{-1/2})/{\sqrt{2}}$$ as the minimum value, both of which turned out to be correct. However, as I decided to solve for $x,y,z$ by the method of substitution, the calculations became somewhat hostile. 

I'm sure there must be a simpler way to arrive at the solutions, and I would be thrilled if someone here would be so generous as to share such a solution. 


We have $$\frac\partial{\partial x}f(x,y,z)=(3-2x(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
$$\frac\partial{\partial y}f(x,y,z)=(4-2y(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
$$\frac\partial{\partial z}f(x,y,z)=(10z-2z(3x+4y+5z^2))e^{-x^2-y^2-z^2}$$
At a stationary point, either $z=0$ and then $3y=4x$, $x=\pm\frac3{10}\sqrt 2 $.
Or $3x+4y+5z^2=5$ and then $x=\frac3{10}$, $y=\frac25$.
