# LaGrange multipliers with natural base

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...

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

whiteway
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.

Homework Helper
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?

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

Homework Helper
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.

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