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.



\nablaf=\lambda\nablag

\nablaf=<2xy,x^{2}>
\nablag=<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.