It really pains me to ask this question, but I was working on an optimization problem and wanted to show a friend how we could also use Lagrange Multipliers to solve it.
I was considering the standard problem of finding the maximum area of a rectangle under a parabola. Let the parabola be given by, $f(x)=4-x^2$ and the area of the rectangle is given by $A(x,y) = 2xy$.
I make a new function and look at it's zero level surface namely, $h(x,y) = y-4+x^2 =0$. Now, $\nabla h(x,y) = (2x,1), \nabla A(x,y) = (y,x)$. I want, $(2x,1) = \xi (y,x)$.\
Can someone tell me where I am going wrong? This is also a result of me using legrange multipliers for the first time.