Maximizing f(x,y) on the circle x^2 + y^2 = 12

[dazedoutpinoy]

Homework Statement

Maximize the function f(x,y) = x^2y constrained by the circle x^2 + y^2 = 12





The attempt at a solution

I already went as far as solving lambda in my work; however, it's still a variable so I could not plug it into solve for x and y.

http://img.photobucket.com/albums/v407/dazedoutpinoy/Calculus001.jpg"

 

[lalbatros]

(it took me a while to understand the meaning of (i) and (j), ...)

From the two equations you have writen, you can now eleminate lambda.
In this way, you will get a curve (two lines actually) where the gradient of f is perpendicular to circles centered on the origin.
(Df = lambda Dg means that as you know: the gradient of f should have no finite component along the given circle)

Then, you should simply find out the intersection of this curve with the particular circle you are targeting.