(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the stationary value of

$$ f(u,v,w) = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m $$

Constraint: $$ u^2 + v^2 + w^2 = t^2 $$

Note: $$ u, v, w > 0 $$. $$ c,d, e, t > 0 $$. $$ m > 0 $$ and is a positive integer.

2. Relevant equations

I have found the auxiliary function:

$$ F = \left( \frac{c}{u} \right)^m + \left( \frac{d}{v} \right)^m + \left( \frac{e}{w} \right)^m - \lambda ( u^2 + v^2 + w^2 - t^2) $$

and

$$ F_u = \frac{- m c^m}{ u^{m+1}} - 2\lambda u = 0 $$

$$ F_v = \frac{- m d^m}{ v^{m+1}} - 2\lambda v = 0 $$

$$ F_w = \frac{- m e^m}{ w^{m+1}} - 2\lambda w = 0 $$

3. The attempt at a solution

Its after this I am having difficult. I don't understand what to do. I assume I have to rearrange the above equations to get $$ u^2, v^2, w^2 $$ and substitute into the constraint, but I can't see how to do that because I get expressions for $$ u^{m + 2} , v^{m+2} , w^{m+2} $$ . Help please.

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Lagrange Multiplier Method

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