Find the number of the points where $f(x,y)=(sx^2+ty^2+2uxy)e^{-x^2-y^2}$ has local maximum or minimum?

Question

Let $s,t,u\in\mathbb R$ satisfy $s>0$, $st>u^2$, $s\neq t$ and define $f$ as $$f(x,y)=(sx^2+ty^2+2uxy)e^{-x^2-y^2}.$$

Find the number of points at which $f$ has local maximum and local minimum, respectively. (I have to find the number of local maximum/minimum, and don't have to calculate the value of $f$ at them.)

I have to find where $f$ has local maximum/minimum, so what I recall is second partial derivative test.

I have $$g_x=2e^{-x^2-y^2}(sx+uy-sx^3-txy^2-2ux^2y)$$ $$g_y=2e^{-x^2-y^2}(ty+ux-ty^3-sx^2y-2uxy^2)$$

If I use second derivative test, I have to calculate $g_x=g_y=0,$ but this is very troublesome.

I am told to find the number of local maximum/minimum points, so I think that there is smarter way other than the second partial derivative test.

Is there any way ?

The similar problem is here Find critical points, minima and maxima of several variable function. Parhaps this helps my question (e.g. by change of variables).

Answer 1

Your function is a product of two terms:

• $e^{-(x^2+y^2)}$ is a radially symmetric gaussian centered at $0$, and
• $sx^2+ty^2+2uxy$ is a paraboloid that has slanted ellipses as level curves.

The condition that $st > u^2$ ensures the level curves are ellipses and not hyperbolas.

If we're thinking about maximums and minimums, the exponential factor has a maximum of $1$ at $(0,0)$, and has no minimum, although it rapidly decreases to zero and its height is bounded below by zero.

For the other factor, let's use the case $u=0$ as a guide: $sx^2+ty^2$ is like a paraboloid but "pinched" closer to one plane, say, the $(y,z)$ plane if $s>t$ (i.e. the level curves are ellipses with the $y$ axis as a major axis). Also note that it has a root at $(0,0)$.

We know the exponential will bring the polynomial down to zero for larger $|(x,y)|$, and in turn the polynomial root at $(0,0)$ will bring the exponential to zero in that region. The minor axes of the paraboloid are closer to the exponential's maximum, so they will have a higher value in the multiplication than the major axes. Therefore, we will have two maximums, one for each side of the hyperbola. Since both terms are positive, there will be no minimum.

The cases where $u$ isn't zero are similar, but the hyperbola is not aligned with the coordinate axes. There is one more sub-case, where $u=0$ and $s=t$, in which case the first factor is a radially symmetric paraboloid and there is a whole circle where the maximum of the function is achieved, but that case is excluded.