I'm a TA for an honors calculus course and I've come upon a homework problem (for which I'm supposed to write solutions!) that has me stumped. This is problem 12, in Chapter 9.13 in volume 2 of Apostol's Calculus book.
Find the critical values of $f(x,y) = (5x+7y-25)e^{-(x^2 + xy + y^2)}$ and classify them as max/min/saddle points.
The back of the book gives the solution as absolute min at $(-\frac{1}{26}, -\frac{3}{26})$ and absolute max at $(1,3)$, so I'm not so interested in just knowing the points.
However, if one sets $f_x = 0 = f_y$ and solves simultaneously, one finds that, say, $y$ satisfies a quartic polynomial (and it's nothing simple like a quadratic polynomial in $y^2$). Now, I'm aware that we do know the quartic formula, but I'm thinking it shouldn't be necessary to use it.
In the paragraph before these exercises (this is problem 12), Apostol says (paraphrased) "When $f(x,y) = e^{1/g(x,y)}$ with $g(x,y) = x^2 + 2 + \cos^2 y -2\cos y$, we can find the critical points via the usual method, but if instead we write $g(x,y) = 1 + x^2 + (1-\cos y)^2$, we see that $f$ has a relative maxima at the points at which $x^2 = 0$ and $(1-\cos y)^2 = 0$".
I mention this because I think this particular exercise should be solved via a clever trick as in the previous paragraph, but I've not been able to come up with the trick on my own.
Is there a nice trick for finding the critical points?
Thank you!