Lagrange multipliers problem

[glid02]

Hey, I need help with a problem involving Lagrange multipliers...

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

 

[HallsofIvy]

Hey, I need help with a problem involving Lagrange multipliers...

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

$$2x \vec{i}- 2y\vec{j}= \lambda 2x\vec{i}+ \lambda 2y\vec{j}$$
Dont forget that these are vector so that your next equations are true:

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.

Okay, good. x= lambda x gives immediately that either x= 0 or lamba= 1. If x= 0, then, since x^2+y^2=289, y^2= 289, y= 17 or -17.
If lambda= 1, y must be 0 so x^2= 289 and x= 17 or -17.

y= -lambda y gives immediately that either

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

I'm not at all sure why you are worried about listing them "lexicographically". If the problem just asks you to find maximum and minimum, then it doesn't matter how you order them.

To answer your question about "why do both the lexicographically first points contain negative coordinates?" (which has nothing to do with "maximization" or "Lagrange"), remember that "lexicographic" ordering is derived from alphabetical ordering of words: look at the first letter and order by that. If both words have the same first letter, look at the second letter and so on. In "lexicographic" ordering of ordered pairs of numbers, look at the first number and order numerically by that. Of course, pairs having first number negative will come before pairs having first number positive. If both first numbers are the same (both 0 for example), then order by the second member. Again, negative numbers are "less than" positive numbers and will come first.