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 49 . (Simplify your answer.)
B. There is no absolute maximum value.
What is the absolute minimum value? Select the correct choice below and, if necessary, fill in the answer box to
complete your choice.
A. The absolute minimum value is . (Simplify your answer.)
B. There is no absolute minimum value.

Ask by Griffiths Wade. in the United States

Mar 21,2025

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 49 . (Simplify your answer.)
B. There is no absolute maximum value.
What is the absolute minimum value? Select the correct choice below and, if necessary, fill in the answer box to
complete your choice.
A. The absolute minimum value is . (Simplify your answer.)
B. There is no absolute minimum value.

Ask by Griffiths Wade. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

We start by rewriting the function:
$$
f(x,y) = x^2 + y^2 - 4y + 4
$$
Complete the square in $$ y $$ by grouping:
$$
y^2 - 4y + 4 = (y-2)^2.
$$
Thus, the function can be written as:
$$
f(x,y) = x^2 + (y-2)^2.
$$

This shows that $$ f(x,y) $$ is the squared distance from the point $$(x,y)$$ to the point $$(0,2)$$.

The region is given by:
$$
R = \{(x,y) : x^2 + y^2 \leq 25\},
$$
which is a disk of radius 5 centered at the origin.

**Absolute Minimum:**

Since $$ f(x,y) \geq 0 $$ (being a sum of squares), the minimum value occurs when the squared distance is as small as possible. The distance from $$(x,y)$$ to $$(0,2)$$ is zero at the point:
$$
(0,2).
$$
Check that $$(0,2)$$ is in $$ R $$:
$$
0^2 + 2^2 = 4 \leq 25.
$$
Thus, the absolute minimum value is:
$$
f(0,2) = 0^2 + (2-2)^2 = 0.
$$

**Absolute Maximum:**

The absolute maximum occurs where the squared distance from $$(0,2)$$ is the largest while still remaining on or within the disk $$ R $$. Notice that on the boundary, we have:
$$
x^2 + y^2 = 25.
$$
The maximum distance from $$(0,2)$$ within $$ R $$ is the distance between the point $$(0,2)$$ and the point on the boundary farthest from $$(0,2)$$.

The distance between the center of the disk $$(0,0)$$ and $$(0,2)$$ is:
$$
\| (0,2) \| = 2.
$$
Thus, the farthest distance from $$(0,2)$$ to any point on the disk is:
$$
2 + 5 = 7.
$$
Therefore, the function reaches its maximum at a point on the boundary such that the distance is $$ 7 $$. Hence, the absolute maximum value is:
$$
f(x,y) = 7^2 = 49.
$$

**Final Answers:**

- The absolute maximum value is 49.
- The absolute minimum value is 0.
The absolute maximum value is 49, and the absolute minimum value is 0.

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