Local Max/Min and saddle points

In summary, the function f(x,y) = (x-y)(1-xy) has saddle points at (1,1) and (-1,-1). This can be determined by finding the partial derivatives of the function and solving the equations simultaneously using elimination.
  • #1
ahhppull
62
0

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.
 
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  • #2
ahhppull said:

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
 
  • #3
Oh, I managed to type out the whole fy wrong.

It is -x^2 -1 +2yx
 
  • #4
ahhppull said:
Oh, I managed to type out the whole fy wrong.

It is -x^2 -1 +2yx
So, can you solve the problem now?
 
  • #5
SammyS said:
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.
 
  • #6
Solve these equations simultaneously:

1-2xy +y2 = 0

-x2 -1 +2yx = 0

Use elimination: add them together.
 
  • #7
SammyS said:
Solve these equations simultaneously:

1-2xy +y2 = 0

-x2 -1 +2yx = 0

Use elimination: add them together.

Thanks man, figured it out
 
  • #8
ahhppull said:
Thanks man, figured it out

So what solution or solutions do you get?

RGV
 
  • #9
Ray Vickson said:
So what solution or solutions do you get?

RGV

Oh, I got saddle points at (1,1) and (-1,-1)
 
  • #10
ahhppull said:
Oh, I got saddle points at (1,1) and (-1,-1)

Correct.

RGV
 

Related to Local Max/Min and saddle points

1. What is a local maximum/minimum point?

A local maximum/minimum point is a point on a graph where the value of the function is greater/smaller than the values of all nearby points. It is also known as a turning point because the function changes direction at this point.

2. How do you find a local maximum/minimum point?

To find a local maximum/minimum point, you need to calculate the first derivative of the function and set it equal to zero. Then, solve for the variable to find the critical points. Next, use the second derivative test to determine whether the critical point is a local maximum, local minimum, or neither.

3. What is a saddle point?

A saddle point is a point on a graph where the tangent plane is horizontal in one direction and vertical in the other direction. This means that the point is not a local maximum or minimum, but rather a point of inflection where the function changes concavity.

4. How do you identify a saddle point?

To identify a saddle point, you need to calculate the second derivative of the function. If the second derivative is equal to zero at a critical point, then it is a potential saddle point. You can then use the second derivative test to confirm whether or not it is a saddle point.

5. What is the significance of local maxima/minima and saddle points in real-world applications?

Local maxima and minima are important in optimization problems, where you want to find the maximum or minimum value of a function. Saddle points are important in determining the stability of a system. In economics, for example, saddle points can represent equilibrium points where the supply and demand curves intersect, indicating a stable market.

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