Re: maximization problem

On Jun 5, 6:58 pm, "Stephen J. Herschkorn" <sjhersc...@xxxxxxxxxxxx>
wrote:
chri...@xxxxxxxxx wrote:
Find the maximum of f(x1, ..., x_n) = (x1*x2*...*x_n)^2 subject to the
constraint

g(x1,...,x_n) = x1^2 + ... + x_n^2 = 1

I try to use lagrange multipliers. I take grad(f) and grad(g) and set

grad(f) = \lambda*grad(g)

but I always get stuck solving the nonlinear system of equations. Is
there a trick to doing this?

Letting y_i = (x_i)^2, your problem is

max sum(i=1..n, ln y_i)
s.t.
sum(i=1..n, y_i) = 1
y_i > 0 for all i

I suspect one could come up with some non-Lagrange argument that y_i =
1/n for all i should be the answer.

Okay, so f(y1, ..., y_n) = ln(y1*...*y_n) = ln(y1) + ... + ln(y_n).
g(y1,...,y_n) = y1 + ... +y_n = 1

Taking the gradients I get the vector equation:

(1/y1, ..., 1/y_n) = \lambda*(y1, ..., y_n)

I dot product both sides with (y1, ..., y_n) to get:

n = \lambda*||y||^2

\lambda = n / ||y||^2

Plugging lambda back in I get

(1/y1, ..., 1/y_n) = n / ||y||^2*(y1, ..., y_n)

So 1 / y_i = n / ||y||^2*y_i

How did you get y_i = 1/n for all i from that?



--
Stephen J. Herschkorn sjhersc...@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan


.

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