Local Max/Min and saddle points

[ahhppull]

Homework Statement

Find the local max/min or saddle points of f(x,y) = (x-y)(1-xy)

Homework Equations

The Attempt at a Solution

I expanded the equation to f(x,y) = x-y-(x^2)y+xy^2.

Then I found the partial derivatives of the function.
fx = 1-2xy +y^2
fy = -x^2-2xy

I'm stuck after this part. Usually I can set the function to 0 and solve for x or y, but I can't do that here.

 

[Ray Vickson]

Homework Statement

Find the local max/min or saddle points of f(x,y) = (x-y)(1-xy)

Homework Equations

The Attempt at a Solution

I expanded the equation to f(x,y) = x-y-(x^2)y+xy^2.

Then I found the partial derivatives of the function.
fx = 1-2xy +y^2
fy = -x^2-2xy

I'm stuck after this part. Usually I can set the function to 0 and solve for x or y, but I can't do that here.

Your fy is incorrect.

RGV

 

[ahhppull]

Oh, I managed to type out the whole fy wrong.

It is -x^2 -1 +2yx

 

[SammyS]

Oh, I managed to type out the whole fy wrong.

It is -x^2 -1 +2yx

So, can you solve the problem now?

 

[ahhppull]

So, can you solve the problem now?

No. I don't know how to solve for 0. I can't set both y or x on either side of the equation.

 

[SammyS]

Solve these equations simultaneously:

1-2xy +y² = 0

-x² -1 +2yx = 0

Use elimination: add them together.

 

[ahhppull]

Solve these equations simultaneously:

1-2xy +y² = 0

-x² -1 +2yx = 0

Use elimination: add them together.

Thanks man, figured it out

 

[Ray Vickson]

Thanks man, figured it out

So what solution or solutions do you get?

RGV

 

[ahhppull]

So what solution or solutions do you get?

RGV

Oh, I got saddle points at (1,1) and (-1,-1)

 

[Ray Vickson]

Oh, I got saddle points at (1,1) and (-1,-1)

Correct.

RGV