Maximising a function with constraints

[tcheret]

I am trying to find the maximun of the following dual variable function f(x,y) = a*(x to the power b) (y to the power c), where a, b and c are constant
with the following constraint g = d*(x to the power e) (y to the power f), where d, e and f are constant

I tried the Lagrangian method without success. Any suggestion on the method i should use?

 

[tkhunny]

Let's see your Lagrange work.

Have you considered there might not be a solution of the type you seek?

 

[Steven G]

I will type out the function so it is a bit clearer for others to see.

f(x,y)= ax^by^c with constraint g= dx^ey^f

 

[tcheret]

Let's see your Lagrange work.

Have you considered there might not be a solution of the type you seek?

I will type out the function so it is a bit clearer for others to see.

f(x,y)= ax^by^c with constraint g= dx^ey^f

That is correct.
Those are exponential functions, so the maximun depend on the exponant.

 

[Deleted member 4993]

That is correct. Those are exponential functions, so the maximun depend on the exponant.

As requested in response #2

Let's see your Lagrange work.

 

[HallsofIvy]

No, they are NOT "exponential functions" because the exponents are constants not variables. But the maximum will depend on all three of a, b, and c. Now, please do as you were asked- show what you did trying to use the "Lagrange multiplier method" and why it did not work..

 

[Deleted member 4993]

That is correct. Those are exponential functions, so the maximum depend on the exponent.

As indicated in response #6, those are NOT exponential functions - those are Power functions.

Also, avoid using 'e' for an indeterminate constant (as in x to the power e) - e like \(\displaystyle \pi\) has special value (~2.718282...)

 

[tcheret]

No, they are NOT "exponential functions" because the exponents are constants not variables. But the maximum will depend on all three of a, b, and c. Now, please do as you were asked- show what you did trying to use the "Lagrange multiplier method" and why it did not work..

See my draft in attachment. The Lagrangian end up with a condition on the exposant bf = ce and a relationship between x and y.
Another way to consider the problem is that f and are 2 surfaces and we are looking for their intersection. They can either intersect or not.(i not sure if can intersect on a single point). If they intersect, they intersect on a line and the maximum is the maximum, which could be infinity.
So, could we interpret the Lagrangian result has followed? bf=ce is the condition for the surface for intercept? it seems to be a very strong condition.
and the relationship between x and y is describing the intersection line?
Any help is welcome.

 

[tcheret]

Also this problem was simplified to 2 variables: x and y. Is there a general resolution for a higher number of variable?