Question
Find all critical points of the following function.
What are the critical points? Select the correct choice below and fill in any answer boxes within your choice.
A. The critical point(s) is/are
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no critical points.
What are the critical points? Select the correct choice below and fill in any answer boxes within your choice.
A. The critical point(s) is/are
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There are no critical points.
Ask by Burton Matthews. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The critical points are
Solution
We start with
First, we find the partial derivatives of
- The partial derivative with respect to
is
- The partial derivative with respect to
is
Critical points occur where both partial derivatives are zero. So, we set up the equations:
From the first equation, solve for
Substitute
Simplify the square:
Multiply the entire equation by 81 to eliminate the fraction:
Factor out
This gives two possibilities:
-
.
Substitute back into: So one critical point is. -
which simplifies to: The real solution is:Substituteinto : So the other critical point is.
Thus, the correct choice is:
A. The critical point(s) is/are
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Explain 
Simplify this solution Bonus Knowledge
To find the critical points of the function
The partial derivative with respect to
And the partial derivative with respect to
Setting
Setting
From equation (1), we can express
Substituting this into equation (2):
This simplifies to:
Multiplying through by
Factoring out
This gives us:
For
Substituting
So one critical point is
For
Substituting
Thus the second critical point is
The critical points are:
A. The critical point(s) is/are (0, 0), (9, -9).
