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.

Ask by Robbins Huff. 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
B. (Simplify your answer.)
B. There is no absolute maximum value.

Ask by Robbins Huff. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

1. Rewrite the function by completing the square in $$ y $$:
   $$
   f(x,y)=x^2+y^2-4y+4=x^2+(y^2-4y+4)=x^2+(y-2)^2.
   $$
   Thus, 
   $$
   f(x,y)=x^2+(y-2)^2,
   $$
   which represents the squared distance from $$(x,y)$$ to the point $$(0,2)$$.

2. The region $$ R $$ is given by:
   $$
   R=\{(x,y):x^2+y^2\leq25\},
   $$
   which is the closed disk of radius 5 centered at the origin $$(0,0)$$.

3. The absolute maximum value of $$ f(x,y) $$ on $$ R $$ occurs at the point in $$ R $$ that is farthest from $$(0,2)$$. Geometrically, the distance from any point $$(x,y)$$ in $$ R $$ to $$(0,2)$$ is maximized when the point is on the boundary $$ x^2+y^2=25 $$ and lies along the line joining $$(0,2)$$ and the center $$(0,0)$$.

4. The distance between the centers $$(0,0)$$ and $$(0,2)$$ is:
   $$
   \text{Distance} = 2.
   $$
   Moving from $$(0,2)$$ in the opposite direction of $$(0,0)$$ a distance equal to the radius (5) reaches the farthest point in $$ R $$.

5. Therefore, the maximum distance from $$(0,2)$$ is:
   $$
   2+5=7,
   $$
   and the maximum value of the function is the square of this distance:
   $$
   f_{\text{max}}=7^2=49.
   $$

6. The absolute maximum value on $$ R $$ is $$ 49 $$.

A. The absolute maximum value is $$ 49 $$.
The absolute maximum value is 49.

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