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.
So, can you solve the problem now?ahhppull said:Oh, I managed to type out the whole fy wrong.
It is -x^2 -1 +2yx
SammyS said:So, can you solve the problem now?
SammyS said:Solve these equations simultaneously:
1-2xy +y2 = 0
-x2 -1 +2yx = 0
Use elimination: add them together.
ahhppull said:Thanks man, figured it out
Ray Vickson said:So what solution or solutions do you get?
RGV
ahhppull said:Oh, I got saddle points at (1,1) and (-1,-1)
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.
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.
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.
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.
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.