Solving Lagrange Multipliers: Max/Min f(x,y)

[ForMyThunder]

Homework Statement

Using Lagrange multipliers, find the maximum and minimum values of f(x,y)=x^3y with the constraint 3x^4+y^4=1.

Homework Equations

The Attempt at a Solution

Here is my complete solution. I just wanted to make sure there are no errors and I did it correctly. Thanks for any feedback.

\nabla f = \lambda \nabla g
3x^2y=12\lambda x^3 and x^3=4\lambda y^3
Solving these I got \lambda = \frac{1}{4} and \lambda = -\frac{1}{4}

Putting these values into the equation on the right gives x=y and x=-y. Substituting these into the left equation gives x=y=\frac{1}{\sqrt{2}} and x=\frac{1}{\sqrt{2}}, y = -\frac{1}{\sqrt{2}}.

Putting these values into the equation for f gives a maximum of \frac{1}{4} and minimum of -\frac{1}{4}.

 

[arildno]

Your maximum and minimum values are correct, but you have, for the maxima two soluitons,
x=y=\pm\frac{1}{\sqrt{2}}
rather then just one maximum.

Similarly for the minimum value.