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ForMyThunder
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Homework Statement
Using Lagrange multipliers, find the maximum and minimum values of [tex]f(x,y)=x^3y[/tex] with the constraint [tex]3x^4+y^4=1[/tex].
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.
[tex]\nabla f = \lambda \nabla g[/tex]
[tex]3x^2y=12\lambda x^3[/tex] and [tex]x^3=4\lambda y^3[/tex]
Solving these I got [tex]\lambda = \frac{1}{4}[/tex] and [tex]\lambda = -\frac{1}{4}[/tex]
Putting these values into the equation on the right gives x=y and x=-y. Substituting these into the left equation gives [tex]x=y=\frac{1}{\sqrt{2}}[/tex] and [tex]x=\frac{1}{\sqrt{2}}[/tex], [tex]y = -\frac{1}{\sqrt{2}}[/tex].
Putting these values into the equation for [tex]f[/tex] gives a maximum of [tex]\frac{1}{4}[/tex] and minimum of [tex]-\frac{1}{4}[/tex].
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