<s>[INST] Min and max of $f(x,y)=e^{-xy}$ where $x^2+4y^2 \leq 5$ I am trying to use Lagrange multipliers to find the maximum and minimum values of the function $$f(x,y)=e^{-xy}$$
constrained as $$x^2+4y^2=5$$
I began this problem by setting up the Lagrangian:
$$f(x,y) = e^{-xy}$$
$$g(x,y) = x^2+4y^2-5$$
$$L(x,y) = f(x,y) - \lambda g(x,y) = e^{-xy} - \lambda (x^2+4y^2-5)$$
So our equations are:
$$-ye^{-xy} - 2\lambda x = 0$$
$$-xe^{xy} -8\lambda y = 0$$
$$x^2+4y^2-5 = 0$$
Now from the first equation, $e^{-xy} = 2\lambda x/y$. Substituting into the second equation yields: $2x^2 \lambda / y = 8 \lambda y$ or $x^2+4y^2 = 0$. This clearly violates the third equation, meaning this system has no solution. Doe this mean there are no local maxima or minima?
Any guidance is greatly appreciated!
 [/INST] The function $ \ f(x \ , \ y) \ = \ e^{\ -xy} \ $ to be extremized has "diagonal symmetry", which is to say that $ \ f(-x \ , \ -y) \ = \ f(x \ , \ y) \ \ ; $ it also has an anti-symmetric property in that $ \ f(-x \ , \ y) \ = \ f(x \ , \ -y) \ = \ \frac{1}{f(x \ , \ y)} \  \ . $  The elliptical region within which we wish to locate extrema, $ \ x^2 + 4y^2 \ \le \ 5 \ \ , $ has "four-fold" symmetry about the origin, so we expect to find the minima of $ \ f(x \ , \ y) \ $ in the first and third quadrants at the points $ \ ( \pm x \ , \ \pm y ) \ $ and its maxima in the second and fourth quadrants at  $ \ ( \pm x \ , \ \mp y ) \ \ . $
While we can perform the usual critical-point analysis in this elliptical region, we may also examine the values of our function on the concentric ellipses $ \ x^2 + 4y^2 \ = \ c^2 \ \ , \ $  with $ \ 0 \ \le \ c^2 \ \le \ 5 \ $ to find the value of $ \ c \ $ where the extrema reach their greatest and smallest values.  Due to the symmetries noted, we can just look at each ellipse in the first quadrant, for which  $ \ f(x \ , \ y) \ = \ \large{e^{\ -\sqrt{c^2 - 4y^2} \ · \ y }}  \ \ . $  This plainly takes on its largest value at the origin, where $ \ c \ = \ 0 \ \Rightarrow \ f(0 \ , \ 0) \ = \ e^{\ 0} \ = \ 1 \ \ . $  We have, for $ \ c \ \neq \ 0 \ \ , $ the derivative
$$  \frac{d}{dy}  \ [ \ e^{\ -\sqrt{c^2 - 4y^2} \ · \ y} \ ]  \ \ =  \ \ e^{\ -\sqrt{c^2 - 4y^2} \ · \ y} \ · \ \frac{d}{dy} \ [ \ -\sqrt{c^2 - 4y^2} \ · \ y \ ]   $$
and since $ \ f(x \ , \ y) \ \neq \ 0 \ \ $ for real values of the coordinate variables, we find the minimum value of our function from
$$ \frac{d}{dy} \ [ \ -\sqrt{c^2 - 4y^2} \ · \ y \ ] \ \ = \ \ -\frac12·\frac{1}{\sqrt{c^2 - 4y^2}}·(-8y) \ · \ y \ + \ -\sqrt{c^2 - 4y^2} \ · \ 1 \ \ = \ \ 0  $$
$$ \Rightarrow \ \ -4y^2 \ \ = \ \ c^2 - 4y^2 \ \ \Rightarrow \ \ y^2 \ \ = \ \ \frac{c^2}{8} \ \ \Rightarrow \ \ x^2 \ \ = \ \ c^2 \ - \ 4·\left(\frac{c^2}{8} \right) \ \ = \ \ \frac{c^2}{2} \ \ . $$
The minima on the concentric ellipses are therefore $ \ f\left(\pm \sqrt{\frac{c^2}{2}} \ , \ \pm \sqrt{\frac{c^2}{8}} \right) \  = \ \large{e^{\ -  \sqrt{\frac{c^2}{2}} \ · \ \sqrt{\frac{c^2}{8}} }} \   =   \ \normalsize{e^{\ -  c^2/4} } \  \le  \ 1 \ \  $ and the maxima are
$ \ f\left(\pm \sqrt{\frac{c^2}{2}} \ , \ \mp \sqrt{\frac{c^2}{8}} \right) \ = \ \large{e^{\ \sqrt{\frac{c^2}{2}} \ · \ \sqrt{\frac{c^2}{8}} }} \   =   \ \normalsize{e^{\ c^2/4} } \ \ge \ 1 \ \ .   $
The global extrema are attained by making $ \ c^2 \ $ as large as is permitted on the elliptical region, that is, at its boundary:
$$ \mathbf{\text{global minima:}} \ \ \ f\left(\pm \sqrt{\frac{5}{2}} \ , \ \pm \sqrt{\frac{5}{8}} \right) \ \ = \ \ e^{\ - 5/4}   \ \ ; $$
$$ \mathbf{\text{global maxima:}} \ \ \ f\left(\pm \sqrt{\frac{5}{2}} \ , \ \mp \sqrt{\frac{5}{8}} \right) \ \ = \ \ e^{\ 5/4}   \ \ . $$
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