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

In summary, using Lagrange multipliers, the maximum and minimum values of f(x,y)=x^3y with the constraint 3x^4+y^4=1 are \frac{1}{4} and -\frac{1}{4} respectively.
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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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  • #2
Your maximum and minimum values are correct, but you have, for the maxima two soluitons,
[tex]x=y=\pm\frac{1}{\sqrt{2}}[/tex]
rather then just one maximum.

Similarly for the minimum value.
 

What are Lagrange multipliers?

Lagrange multipliers are a mathematical method used to optimize a function subject to constraints. They allow us to find the maximum or minimum value of a function while satisfying a set of constraints.

When should Lagrange multipliers be used?

Lagrange multipliers should be used when we need to optimize a function subject to constraints that cannot be easily incorporated into the function itself. This method allows us to incorporate the constraints into the optimization process.

How do we solve for Lagrange multipliers?

To solve for Lagrange multipliers, we first set up the Lagrangian function by adding the constraint(s) to the original function. Then, we take the partial derivatives of the Lagrangian function with respect to the variables and set them equal to 0. This will give us a system of equations that can be solved to find the values of the variables and the Lagrange multiplier.

What is the significance of the Lagrange multiplier?

The Lagrange multiplier represents the rate of change of the function with respect to the constraints. It tells us how much the function will change for a small change in the constraints. In other words, it is the amount by which the maximum or minimum value of the function will change when the constraints are relaxed by a small amount.

Are there any limitations to using Lagrange multipliers?

While Lagrange multipliers are a powerful tool for solving optimization problems, they can only be used for single-variable constraints. If the constraints involve multiple variables, then other methods such as the Kuhn-Tucker conditions may need to be used.

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