Question
Find the absolute maximum and minimum values of the following function on the given set
What is the absolute maximum value? Select the correct choice below and, if necessary, fill in the answer box to
complete your choice.
A. The absolute maximum value is
B. (Simplify your answer.)
B. There is no absolute maximum value.
What is the absolute maximum value? Select the correct choice below and, if necessary, fill in the answer box to
complete your choice.
A. The absolute maximum value is
B. (Simplify your answer.)
B. There is no absolute maximum value.
Ask by Robbins Huff. in the United States
Mar 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The absolute maximum value is 49.
Solution
-
Rewrite the function by completing the square in
: Thus,which represents the squared distance fromto the point . -
The region
is given by: which is the closed disk of radius 5 centered at the origin. -
The absolute maximum value of
on occurs at the point in that is farthest from . Geometrically, the distance from any point in to is maximized when the point is on the boundary and lies along the line joining and the center . -
The distance between the centers
and is: Moving fromin the opposite direction of a distance equal to the radius (5) reaches the farthest point in . -
Therefore, the maximum distance from
is: and the maximum value of the function is the square of this distance: -
The absolute maximum value on
is .
A. The absolute maximum value is
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Simplify this solution Bonus Knowledge
To find the absolute maximum and minimum values of the function
Next, we need to determine the critical points of
Next, we check the boundary of
At the boundary, consider
To maximize or minimize
-
When
: -
When
:
Thus, the minimum value occurs at the point
For the maximum value, the calculations show that the function achieves the highest value of
Thus, the absolute maximum value is
A. The absolute maximum value is 49.
