Question

Find the maximum and minimum values of the function f(x,y)=4x^2 +9y^2
subject to xy = 4. Use Lagrange multipliers.

Answer 1

This are I did
{1. fx=8x = gx=λy, 2. fy=18y =gx=λx, 3. xy=4
(8x/18y)=(λy/λx)------->8x^2 =18y^2-------->8x^2 -18y^2=0. I don't know what to do next.

Answer 2

So far, so good. Now include the constraint xy=4
4x^2 = 9y^2
4x^2 = 9(4/x)^2
x^2 = 36/x^2
x^4 = 36
x = ±√6
y = ±2/3 √6
So the minimum f(x,y) = 48

This may seem strange, since the minimum value of 4x^2+9y^2 is clearly 0. But with the constraint xy=4, we see that our collection of ellipses is tangent to the ellipse xy=4 only for the ellipse 4x^2+9y^2 = 48.

A nice discussion of the topic is presented here:

https://tutorial.math.lamar.edu/classes/calciii/lagrangemultipliers.aspx