Max min, Lagrange's multiplier question

[chy1013m1]

Homework Statement

http://www.individual.utoronto.ca/chy1013m1/a42.jpg

Homework Equations

possibly Lagrange's multiplier..

The Attempt at a Solution

treating S = f(x1, x2, ... , x2006) = x1 * 1^1/3 + x2 * 2^1/3 + ... + x2006 * 2006^1/3

and constrain G(x1, x2 ... x2006) = x1 ^ 3/2 + x2 ^ 3/2 + ... + x2006 ^ 3/2 - (2^1/2 / (2006^1/2 * 2007 ^ 1/2)) = 0

then solve G(x...) = 0
gradient(f) = lambda * gradient(G) , which isn't all that clear what to do next.. any hints ?

 

[HallsofIvy]

It isn't clear what to do next? How about completing that equation:
$\nabla S= \lambda \cdot \nabla G$? (S, not f)
What is $\nabla S$? What is $\nabla G$? That should give you 2007 linear equations for x₁, x₂, . . . , x₂₀₀₆ and $\lambda$. Fortunately they are almost all separated and solving just a few should give you the general formula.