Critical Points - Multivariable Calc

In summary, a critical point in multivariable calculus is a point on a function where the partial derivatives are equal to zero or undefined. To find critical points, one needs to take the partial derivatives of the function and set them equal to zero. These points are significant as they can indicate the maximum, minimum, or saddle points of a function and are used in optimization problems. It is possible for a function to have multiple critical points, and their nature can be determined by using the second partial derivatives of the function.
  • #1
Mona1990
13
0
Hi, i was wondering if someone could please help to find and classify the critical points of :
f(x,y) = (x-y)^2

What i know:
I got fx = 2(x-y) and fy = -2(x-y)
and in order to find the critical points we need to solve:

2(x-y) =0
-2(x-y) = 0

so if x =y then the above hold.

where would I go on from here on?

thanks!
 
Last edited:
Physics news on Phys.org
  • #2
Basically anything that runs along the line y=x would contain your critical points. Plug y=x into the hessian matrix and then determine what kind of extrema it is.
 
  • #3
Alright! thanks a lot :D
 

What is a critical point in multivariable calculus?

A critical point in multivariable calculus is a point on a function where the partial derivatives are equal to zero or undefined. It is also known as a stationary point, as the function does not change in any direction at that point.

How do you find critical points in multivariable calculus?

To find critical points in multivariable calculus, you need to take the partial derivatives of the function with respect to each variable and set them equal to zero. Solve the resulting system of equations to find the critical points.

What is the significance of critical points in multivariable calculus?

Critical points are important in multivariable calculus because they can indicate the maximum, minimum, or saddle points of a function. They are also used in optimization problems to find the optimal solution.

Can a function have more than one critical point?

Yes, a function can have multiple critical points. In fact, most functions have multiple critical points in multivariable calculus, except for linear functions and some special cases.

How do you determine the nature of a critical point in multivariable calculus?

The nature of a critical point can be determined by using the second partial derivatives of the function. If the second derivative is positive, the critical point is a local minimum. If it is negative, the critical point is a local maximum. If it is zero, further analysis is needed to determine the nature of the critical point.

Similar threads

Can't find the correct max on this surface
    • Calculus and Beyond Homework Help
Replies
6
Views
854
Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
468
Find the dimensions that will minimize the surface area of a Rectangle
    • Calculus and Beyond Homework Help
Replies
1
Views
461
Solving Multivariate Problem: Critical Points for ##y=-x##
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Critical points of a diff eq system
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Another max value on surface (no boundaries)
    • Calculus and Beyond Homework Help
Replies
8
Views
1K
How should I show that ## B ## is given by the solution of this?
    • Calculus and Beyond Homework Help
Replies
2
Views
462
Stationary points classification using definiteness of the Lagrangian
    • Calculus and Beyond Homework Help
Replies
4
Views
835
Find all relative maxima/minima and saddle points
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
How should I calculate the stationary value of ## S[y] ##?
    • Calculus and Beyond Homework Help
Replies
2
Views
543

Hot Threads

Recent Insights

Back
Top