Critical Points & Extrema of f (x, y)

In summary, a critical point of a function is where the derivative is equal to zero or undefined, indicating a flat or undefined slope. To find critical points, take the derivative and set it equal to zero. An extremum of a function is a maximum or minimum value, either locally or globally. The second derivative test can determine the nature of a critical point, and critical points and extrema are important in understanding the behavior of a function and identifying maximum and minimum values.
  • #1
harpazo
208
16
Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails.

f (x, y) = x^3 + y^3

Solution:

f_x = 3x^2

f_y = 3y^2

f_xx = 6x

f_th = 6y

f_2xy = 0

I set f_x and f_y = 0 and found the critical points to be
(0, 0).

Is this right so far?

d = (6x)(6y) - [0]^2

d = 36xy

d = 36 (0)(0) - 0

d = 0

Since d = 0, the test is inconclusive.

I evaluated f (x, y) at (0, 0) and found the point in space to be (0, 0, 0).

The textbook goes on to say that the test for relative extrema fails and that we have a saddle point here.

1. Why does the test fails here?

2. I am not too sure how to show that we have a saddle point in this case.

Can someone explain this question in simple terms?
 
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  • #2
Harpazo said:
Find the critical points and test for relative extrema. List the critical points for which the Second Partials Test fails.

f (x, y) = x^3 + y^3

Solution:

f_x = 3x^2

f_y = 3y^2

f_xx = 6x

f_th = 6y

f_2xy = 0

I set f_x and f_y = 0 and found the critical points to be
(0, 0).

Is this right so far?

d = (6x)(6y) - [0]^2

d = 36xy

d = 36 (0)(0) - 0

d = 0

Since d = 0, the test is inconclusive.

I evaluated f (x, y) at (0, 0) and found the point in space to be (0, 0, 0).

The textbook goes on to say that the test for relative extrema fails and that we have a saddle point here.

1. Why does the test fails here?

2. I am not too sure how to show that we have a saddle point in this case.

Can someone explain this question in simple terms?

In 3D a saddle point is the equivalent of a stationary point of inflection in 2D. As this function is the sum of two functions which have SPOIs at (0,0) it makes sense that in 3D you would have a saddle point.

As for proving it, you need to show that movement in some directions from that point results in a decrease while movement in other directions from that point results in an increase. That would be enough to show that it is not a maximum or a minimum, and thus is a saddle point.
 

FAQ: Critical Points & Extrema of f (x, y)

What is a critical point of a function?

A critical point of a function is a point where the derivative of the function is equal to zero or does not exist. This means that the slope of the function at that point is either flat or undefined.

How do you find critical points of a function?

To find the critical points of a function, you need to take the derivative of the function with respect to each variable and set them equal to zero. You then solve for the variables to find the critical points.

What is an extremum of a function?

An extremum of a function is a point where the function has either a maximum or minimum value. This can be a local extremum, where the function has the highest or lowest value in a small neighborhood, or a global extremum, where the function has the highest or lowest value over its entire domain.

How do you determine if a critical point is a maximum, minimum, or saddle point?

To determine the nature of a critical point, you can use the second derivative test. If the second derivative is positive at the critical point, it is a minimum. If the second derivative is negative, it is a maximum. If the second derivative is zero, further analysis is needed to determine the nature of the critical point.

What is the significance of critical points and extrema in a function?

Critical points and extrema are important in understanding the behavior of a function. They can help identify where the function is increasing, decreasing, or changing direction. They also provide information about the maximum and minimum values of a function, which can be useful in optimizing or solving problems involving the function.

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