Hey, I need help with a problem involving Lagrange multipliers...(adsbygoogle = window.adsbygoogle || []).push({});

Here is the question:

Find the absolute maximum and minimum of the function

f(x,y) = x^2-y^2 subject to the constraint x^2+y^2=289.

As usual, ignore unneeded answer blanks, and list points in lexicographic order.

I found that the max is 289 and the min is -289.

I took the gradients of each and came up with:

2x-2y = lambda2x+lambda2y

x=lambda(x)

-y=lambda(y)

Here's where I'm stuck, I know lambda = +/-1, but I can't come up with an equation to solve for x or y in terms of y or x. The best idea I can come up with is to solve for lambda from -y=lambda(y), which is just

-y/y and plug it into x=lambda(x), but that doesn't help any when solving for y.

I guess lambda = 1 when y = 0 and lambda = -1 when x = 0.

OK, I found the correct answers, but I could still use an explanation as to how to list them lexicographically.

The correct answer is:

max = 289 at points (-17, 0) and (17,0)

min = -289 at points (0,-17) and (0,17)

When y = 0, lambda = 1 so (lambda(x))^2+0=289

which is x^2=289, x = 17

When x = 0, lambda = -1 so (lambda(y))^2+0=289

which is -y^2=289, y = -17

So why do both the lexicographically first points contain negative coordinates?

I just rambled a whole lot. Basically if someone could give me an explanation for why the correct answer is correct that'd be great.

Thanks a lot,

Gregg

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# Homework Help: Lagrange multipliers problem

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