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

Ask by Evans Wade. in the United States

Mar 21,2025

Question

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

Ask by Evans Wade. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To find the absolute maximum and minimum values of the function $$ f(x, y) = 2x^2 - 2y^2 + 10 $$ on the set $$ R = \{(x, y): x^2 + y^2 \leq 1\} $$, we will follow these steps:

1. **Find the critical points of $$ f $$ inside the region $$ R $$**.
2. **Evaluate $$ f $$ on the boundary of the region $$ R $$**.
3. **Compare the values obtained to determine the absolute maximum and minimum.**

### Step 1: Find Critical Points

To find the critical points, we need to compute the partial derivatives of $$ f $$ and set them to zero.

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

Setting these equal to zero gives us:

$$
4x = 0 \quad \Rightarrow \quad x = 0
$$
$$
-4y = 0 \quad \Rightarrow \quad y = 0
$$

Thus, the only critical point is $$ (0, 0) $$.

### Step 2: Evaluate $$ f $$ on the Boundary

The boundary of the region $$ R $$ is defined by $$ x^2 + y^2 = 1 $$. We can parameterize this using trigonometric functions:

$$
x = \cos(\theta), \quad y = \sin(\theta) \quad \text{for } \theta \in [0, 2\pi]
$$

Substituting into $$ f $$:

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

Now, we need to find the maximum and minimum values of $$ 2\cos(2\theta) + 10 $$.

The maximum value of $$ \cos(2\theta) $$ is 1, and the minimum value is -1. Therefore:

- Maximum: $$ 2(1) + 10 = 12 $$
- Minimum: $$ 2(-1) + 10 = 8 $$

### Step 3: Compare Values

Now we compare the values at the critical point and the boundary:

- At the critical point $$ (0, 0) $$:
  $$
  f(0, 0) = 2(0)^2 - 2(0)^2 + 10 = 10
  $$

- On the boundary:
  - Maximum: $$ 12 $$ at points $$ (1, 0) $$ and $$ (-1, 0) $$
  - Minimum: $$ 8 $$

### Conclusion

- The absolute maximum value is $$ 12 $$ and occurs at $$ (1, 0) $$ and $$ (-1, 0) $$.
- The absolute minimum value is $$ 8 $$ and occurs at $$ (0, 1) $$ and $$ (0, -1) $$.

### Final Answers

**Absolute Maximum:**
A. The absolute maximum value is 12 and occurs at $$ (1,0),(-1,0) $$.

**Absolute Minimum:**
A. The absolute minimum value is 8 and occurs at $$ (0,1),(0,-1) $$.
The absolute maximum value is 12 and occurs at (1,0) and (-1,0). The absolute minimum value is 8 and occurs at (0,1) and (0,-1).

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