Find the maximum and minimum values of $f(x,y)=xy$
subject to $\frac{y^2}{2}+\frac{x^2}{4}=1$
Write the Lagrange function in the form $f(x,y) = \lambda g(x,y)$
So far I've got $x=\sqrt2$ or $2\sqrt2$ and $y=1,-1$ or $y=2,-2$
I'm kinda stuck from here and not sure what to do next, can anyone please help me with the question?
Best Answer
Plug the $ (x,y)$ pairs into the function and see which one gives the maximum. It's trial and error at this point.
A little more commentary:
The Lagrange multiplier method gives the condition for an $(x,y)$ point to be maximum or minimum. Once you got this set of points, you have to search among the points to see which one is the one which is helpful in the objective you want to do.
Actually, there is a shortcut, you can use the second derivatives (similar to the single variable) to classify maximum and minima. See more about it here