# Thread: 3 eqns w/ 3 unknowns: 4y2-2x = 2*lambda*x, 8*xy = 2*lambda*y, x2+y2=1

1. ## 3 eqns w/ 3 unknowns: 4y2-2x = 2*lambda*x, 8*xy = 2*lambda*y, x2+y2=1

Hi everyone!

I've been stuck on this one for some hours now (about 5 hours to be honest) and I just can't solve it. It is actually a "find min. and max. points"-question but I've come so far that I just have to solve for the Lagrange multiplier and the three equations that I have are:
4y2-2x = 2*lambda*x
8*xy = 2*lambda*y
x2+y2=1

I need to solve for x and y and x has to be greater than zero.
I have to do this by hand and I really hope that some of you smart people can figure this out! the result should be three points.
Thanks in advance, I appreciate every help I can get!

EDIT: the function that I've initially been given is 4xy2-x2 and it is in the domain of R2 with the restriction: x2+y<=1, x>=0

2. Originally Posted by danishkid
I've been stuck on this one for some hours now (about 5 hours to be honest) and I just can't solve it. It is actually a "find min. and max. points"-question but I've come so far that I just have to solve for the Lagrange multiplier and the three equations that I have are:

4y2-2x = 2*lambda*x
8*xy = 2*lambda*y
x2+y2=1

I need to solve for x and y and x has to be greater than zero.
Try using what you learned back in algebra! I'll replace "lambda" with "z", for easy of typesetting. This means we have:

. . . . .$4y^2\, -\, 2x\, =\, 2xz$

. . . . .$8xy\, =\, 2zy$

. . . . .$x^2\, +\, y^2\, =\, 1$

Solving the first equation for "z=", we get:

. . . . .$\dfrac{2y^2}{x}\, -\, 1\, =\, z$

Solve the second equation for "z=", we get:

. . . . .$4x\, =\, z$

Putting these together, we get:

. . . . .$4x\, =\, \dfrac{2y^2}{x}\, -\, 1$

. . . . .$4x\, +\, 1\, =\, \dfrac{2y^2}{x}$

. . . . .$2x^2\, +\, \dfrac{x}{2}\, =\, y^2$

Solving the third of the original equations, we get:

. . . . .$y^2\, =\, 1\, -\, x^2$

Putting these together, we get:

. . . . .$2x^2\, +\, \dfrac{x}{2}\, =\, 1\, -\, x^2$

What did you get when you solved this quadratic equation? Where did this lead?

3. Originally Posted by danishkid
Hi everyone!

I've been stuck on this one for some hours now (about 5 hours to be honest) and I just can't solve it. It is actually a "find min. and max. points"-question but I've come so far that I just have to solve for the Lagrange multiplier and the three equations that I have are:
4y2-2x = 2*lambda*x
8*xy = 2*lambda*y
x2+y2=1 HOW DO YOU GET THIS?

I need to solve for x and y and x has to be greater than zero.
I have to do this by hand and I really hope that some of you smart people can figure this out! the result should be three points.
Thanks in advance, I appreciate every help I can get!

EDIT: the function that I've initially been given is 4xy2-x2 and it is in the domain of R2 with the restriction: x2+y<=1, x>=0
I want to start from the original problem.

First, did you give us the actual constraint or is it $x^2 + y^2 \le 1.$

Second, why did you ignore the second constraint of $x \ge 0$?

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