LaGrange multipliers with natural base

• whiteway
In summary, the given equations lead to a graph that is unbounded, meaning there is no minimum value for the function f(x,y,z)=exy. However, the maximum value can be found at (x,y)=(2,2) where f(x,y,z)=e^4.
whiteway

Homework Statement

f(x,y,z)=exy and x5+y5=64

Find Max and Min

Homework Equations

∇F = <yexy, xexy>
λ∇G = <5x4λ, 5y4λ>

The Attempt at a Solution

yexy = 5x4λ
xexy = 5y4λ
x5+y5=64

No idea where to go from here...

I would start by dividing the first equation by the second.

Alright, dividing the first by the second, I get y/x = y4/x^4, or y=x

plugging that in, we get 2x5=64, or x=2

since x=2, y must equal 2

so one solution is e^4, and that was correct as the maximum, but I am having trouble finding the minimum.

I really appreciate the help.

I get y/x=(x/y)^4. Which is a little different. That should warn you to be concerned about the cases where x=0 or y=0. They don't work here. Still it's still something to think about in general. But have you sketched a graph of x^5+y^5=64? It's unbounded. You can get a greatest lower bound for exp(xy), but does it have a minimum?

Ok, so since the graph is unbounded, then there is no minimum?

No, the graph can be unbounded in general and you can have a minimum. But what happens in this case? Tell me why there isn't a minimum.

well, the function can approach e-infinity, or 0, correct?

whiteway said:
well, the function can approach e-infinity, or 0, correct?

Exactly. But only as x->infinity. There is no point where it actually reaches 0. So I would say 'no minimum'. I was sort of torn about this for a while.

Awesome, that makes sense. I really appreciate it. Thank you

1. What is the concept behind LaGrange multipliers with natural base?

LaGrange multipliers with natural base is a mathematical optimization technique used to find the maximum or minimum value of a function, subject to certain constraints. It involves using the LaGrange multiplier method and the natural logarithm function to solve for the optimal values of the variables.

2. How does the LaGrange multiplier method work?

The LaGrange multiplier method involves creating a new function, known as the LaGrange function, by adding the product of the constraint equations and Lagrange multipliers to the original objective function. The optimal values of the variables are then found by setting the partial derivatives of the LaGrange function with respect to each variable and the Lagrange multipliers equal to zero.

3. What is the advantage of using natural base in LaGrange multipliers?

Using natural base in LaGrange multipliers allows for simpler calculations and a more intuitive understanding of the problem. It also eliminates the need for additional constants in the LaGrange function, making the process more streamlined and efficient.

4. Can LaGrange multipliers with natural base be applied to any type of optimization problem?

Yes, LaGrange multipliers with natural base can be applied to any type of optimization problem, as long as the objective function and constraints are continuous and differentiable.

5. Are there any limitations to using LaGrange multipliers with natural base?

One limitation of LaGrange multipliers with natural base is that it may not always provide the global optimal solution, but rather a local optimal solution. Additionally, the method may become more complex when dealing with a large number of constraints or variables.

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