Solving Lagrange Multipliers for f(x,y)=x^{2}y

[SelHype]

Use Lagrange Multipliers to find the maximum and minimum values of f(x,y)=x$$^{2}$$y subject to the constraint g(x,y)=x$$^{2}$$+y$$^{2}$$=1.



$$\nabla$$f=$$\lambda$$$$\nabla$$g

$$\nabla$$f=<2xy,x$$^{2}$$>
$$\nabla$$g=<2x,2y>

1: 2xy=2x$$\lambda$$ ends up being y=$$\lambda$$
2: x$$^{2}$$=2y$$\lambda$$ ends up being(1 into 2) x=$$\sqrt{2\lambda$$ ^{2}}[/tex]
3: x$$^{2}$$+y$$^{2}$$=1

1 and 2 into 3:
(2$$\lambda$$$$^{2}$$)+($$\lambda$$$$^{2}$$)=1

Do I then solve for $$\lambda$$ and x and y? Did I do the above correctly? I am going off of an example of f(x,y,z) so I'm not sure if I'm correct. Any help is greatly appreciated!

 

[µ³]

Be careful when dividing by a variable because x=0 could easily be a solution to this problem. Instead of trying to solve for lambda, try eliminating it. You could try multiplying the first equation by y and the second equation by x. You should get an equation for x and y, and so coupled with equation 3, you have two equations two unknowns.