1 If possible, find the absolute maximum and minimum values of the following function on the set .

Find the absolute maximum value on the set. Select the correct choice below and, if necessary, fill in the answer boxes
within your choice.
A. The absolute maximum value is and occurs at
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There is no absolute maximum value.

Ask by Simmons Savage. in the United States

Mar 21,2025

Question

1 If possible, find the absolute maximum and minimum values of the following function on the set .

Find the absolute maximum value on the set. Select the correct choice below and, if necessary, fill in the answer boxes
within your choice.
A. The absolute maximum value is and occurs at
(Type an ordered pair. Use a comma to separate answers as needed.)
B. There is no absolute maximum value.

Ask by Simmons Savage. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

First, we note that

$$
f(x,y)=2x^2-2y^2+10
$$

and the region is

$$
R=\{(x,y) : x^2+y^2\le 1\}.
$$

### Step 1. Find Critical Points in the Interior

Compute the partial derivatives:

$$
f_x= \frac{\partial f}{\partial x} = 4x, \quad f_y= \frac{\partial f}{\partial y} = -4y.
$$

Set them equal to zero:

$$
4x=0\quad \Longrightarrow \quad x=0,
$$
$$
-4y=0\quad \Longrightarrow \quad y=0.
$$

Thus, the only interior critical point is $$(0,0)$$ and its function value is

$$
f(0,0)=2(0)^2-2(0)^2+10=10.
$$

### Step 2. Analyze the Boundary

The boundary is given by

$$
x^2+y^2=1.
$$

Parameterize using:

$$
x=\cos\theta,\quad y=\sin\theta.
$$

Substitute into $$f$$:

$$
f(\cos\theta,\sin\theta)=2\cos^2\theta-2\sin^2\theta+10.
$$

Recall the double-angle identity:

$$
\cos(2\theta)=\cos^2\theta-\sin^2\theta.
$$

Thus, the function on the boundary becomes:

$$
f(\cos\theta,\sin\theta)=2\cos(2\theta)+10.
$$

The maximum value on the boundary is achieved when

$$
\cos(2\theta)=1,
$$

which gives

$$
f_{\max}=2(1)+10=12.
$$

### Step 3. Determine Where the Maximum Occurs

The condition $$\cos(2\theta)=1$$ is satisfied when:

$$
2\theta=0 \quad (\text{mod }2\pi) \quad \Longrightarrow \quad \theta=0 \quad \text{or} \quad \theta=\pi.
$$

So, the corresponding points are:

- For $$\theta=0$$: $$(x,y)=(\cos 0, \sin 0)=(1,0)$$.
- For $$\theta=\pi$$: $$(x,y)=(\cos\pi, \sin\pi)=(-1,0)$$.

### Conclusion

The highest value $$f(x,y)=12$$ is reached on the boundary at the points $$(1,0)$$ and $$(-1,0)$$.

Thus, the answer is:

A. The absolute maximum value is $$12$$ and occurs at $$(1,0)$$ and $$(-1,0)$$.
The absolute maximum value is 12 and occurs at the points (1,0) and (-1,0).

Upstudy AI Solution

Answer

Solution

Step 1. Find Critical Points in the Interior

Step 2. Analyze the Boundary

Step 3. Determine Where the Maximum Occurs

Conclusion

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