LaGrange multipliers with natural base

[whiteway]

Homework Statement

f(x,y,z)=e^xy and x⁵+y⁵=64

Find Max and Min

Homework Equations

∇F = <ye^xy, xe^xy>
λ∇G = <5x⁴λ, 5y⁴λ>

The Attempt at a Solution

ye^xy = 5x⁴λ
xe^xy = 5y⁴λ
x⁵+y⁵=64

No idea where to go from here...

 

[Dick]

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.

 

[Dick]

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?

 

[Dick]

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?

 

[Dick]

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.

 

[whiteway]

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