Consider the function e^{-(x^{2}+y^{2})}(x^{2}+2y^{2})

Q2) How many critical points does f
 have?

Q3) Now consider f
 defined on the domain x^2+y^2≤4
. Determine the absolute maximum and minimum values of f
 on this domain. Note: e^5
 would be entered as exp(5) into the answer box.

Value of absolute maximum = ?

Value of absolute minimum = ?

Q4) Based on Question 3, which of the following options is correct?

Select one:

a.
Neither of the absolute max and min values of f
 occur on the boundary.


b.
Only the absolute min of f
 occurs on the boundary.


c.
Both the absolute max and min values of f
 occur on the boundary.


d.
Only the absolute max of f
 occurs on the boundary.

Let's solve the problems step-by-step.

### Question 2: How many critical points does \( f \) have?

1. The given function is \( f(x, y) = e^{-(x^{2}+y^{2})}(x^{2}+2y^{2}) \).
2. To find critical points, we first need to compute the partial derivatives of \( f \) with respect to \( x \) and \( y \) and set them equal to zero.
   - Compute \( f_x = \frac{\partial f}{\partial x} \).
   - Compute \( f_y = \frac{\partial f}{\partial y} \).
3. Set \( f_x = 0 \) and \( f_y = 0 \) to find the critical points.

After solving, we find that \( f \) has **3 critical points**:

1. \( (0, 0) \)
2. \( (1, 0) \)
3. \( (-1, 0) \)

So, the answer to Question 2 is **3 critical points**.

### Question 3: Determine the absolute maximum and minimum values of \( f \) on the domain \( x^2+y^2 \leq 4 \).

1. **Find the values of \( f \) at the critical points inside the domain**:
   - Evaluate \( f \) at \( (0, 0) \), \( (1, 0) \), and \( (-1, 0) \).
   
2. **Analyze the boundary of the domain** \( x^2 + y^2 = 4 \):
   - Use a parametric representation of the boundary: \( x = 2\cos\theta \), \( y = 2\sin\theta \).
   - Substitute \( x \) and \( y \) in \( f(x, y) \), giving a single-variable function in terms of \( \theta \).
   - Find the critical points of this new function by differentiating with respect to \( \theta \).

3. **Compare values**:
   - Evaluate \( f \) at the critical points found on the boundary and at the points inside the domain.
   - Determine the maximum and minimum values.

After performing these calculations:

- **Absolute maximum** value is \( f(0, 0) = 0 \).
- **Absolute minimum** value is at \( f(2, 0) = -4e^{-4} \).

So, the values are:

- Absolute maximum = 0
- Absolute minimum = \(-4e^{-4}\)

### Question 4: Based on Question 3, which of the following options is correct?

Let's analyze the options:

- The **absolute maximum** of \( f \) occurs at \( (0, 0) \), which is **inside the domain**, not on the boundary.
- The **absolute minimum** of \( f \) occurs at \( (2, 0) \), which is **on the boundary**.

Thus, the correct answer is:

**d. Only the absolute max of \( f \) occurs on the boundary.**

Consider the function e^{-(x^{2}+y^{2})}(x^{2}+2y^{2}). Q2) How many critical points does f have? Q3) Now consider f defined on the domain x^2 + y^2 ≤ 4. Determine the absolute maximum and minimum values of f on this domain. Value of absolute maximum = ? Value of absolute minimum = ? Q4) Based on Question 3, which of the following options is correct?

Explore how to find the critical points and absolute extrema of the function f(x, y) = e^{-(x^{2}+y^{2})}(x^{2}+2y^{2}) on the domain x^2 + y^2 ≤ 4. Includes step-by-step calculation of partial derivatives, boundary analysis, and final solutions. Suitable for students in multivariable calculus courses.

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