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

danishkid said:

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.

Click to expand...

Try using what you learned back in algebra! I'll replace "lambda" with "z", for easy of typesetting. This means we have:

. . . . .\(\displaystyle 4y^2\, -\, 2x\, =\, 2xz\)

. . . . .\(\displaystyle 8xy\, =\, 2zy\)

. . . . .\(\displaystyle x^2\, +\, y^2\, =\, 1\)

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

. . . . .\(\displaystyle \dfrac{2y^2}{x}\, -\, 1\, =\, z\)

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

. . . . .\(\displaystyle 4x\, =\, z\)

Putting these together, we get:

. . . . .\(\displaystyle 4x\, =\, \dfrac{2y^2}{x}\, -\, 1\)

. . . . .\(\displaystyle 4x\, +\, 1\, =\, \dfrac{2y^2}{x}\)

. . . . .\(\displaystyle 2x^2\, +\, \dfrac{x}{2}\, =\, y^2\)

Solving the third of the original equations, we get:

. . . . .\(\displaystyle y^2\, =\, 1\, -\, x^2\)

Putting these together, we get:

. . . . .\(\displaystyle 2x^2\, +\, \dfrac{x}{2}\, =\, 1\, -\, x^2\)

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

danishkid said:

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

Click to expand...

I want to start from the original problem.

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

Second, why did you ignore the second constraint of \(\displaystyle x \ge 0\)?