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Find the critical points of the following function. Use the Second Derivative Test to determine (if possible) whether each
critical point corresponds to a local maximum, local minimum, or saddle point. If the Second Derivative Test
is inconclusive, determine the behavior of the function at the critical points.

What are the critical points?
(Type an ordered pair. Use a comma to separate answers as needed.)

Ask by Martin Delgado. in the United States
Mar 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The only critical point is , and it is a saddle point.

Solution

Let .
  1. Find the critical points:
    Compute the first partial derivatives:
    Set them equal to zero:
    Thus, the only critical point is .
  2. Apply the Second Derivative Test:
    Compute the second partial derivatives:
    Calculate the determinant of the Hessian matrix:
    Since , the critical point is a saddle point.
The only critical point is , and it corresponds to a saddle point.

Answered by UpStudy AI and reviewed by a Professional Tutor

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Simplify this solution

Beyond the Answer

To find the critical points of the function , we first compute the partial derivatives:
Setting these derivatives equal to zero gives us the system of equations:
  1. which leads to
  2. which leads to
Thus, the only critical point is .
Next, we apply the Second Derivative Test by calculating the second derivatives:
The determinant is computed as follows:
Since , the Second Derivative Test indicates that the critical point is a saddle point.
So, the critical point is:
(0, 0)

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