We will find the maximum and minimum values of f(x, y) = xy on the ellipse (x^2)/4+ y^2 = 1 (so our constraint function is g(x, y) = (x^2)/4 + y^2 ).
a) Find ∇f and ∇g.
b) You should notice that there are no places on the ellipse where ∇g = 0, so we just need to find the points on the ellipse where ∇f(x, y) = λ∇g(x, y) for some λ. Find the points on the ellipse where fx(x, y) = λgx(x, y) and fy(x, y) = λgy(x, y) for some λ. Often a good strategy to use is to solve for λ in one of the components, and then plug this in for λ in the other component (this eliminates the λ’s and allows us to solve for just x and y). Don’t forget that we want the x and y points where (x^2) /4 + y^2 = 1.
c) Plug the points from the previous step into f(x, y) to find the maximum and minimum (you should have four total points, two of these will be the maximum and two will be the minimum).
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