Based on the information given solve the questions below

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

Q2) How many critical points does f
 have = 5 critical points, (−1,0), (0,−1), (0,0), (0,1) (1,0).

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 = 2exp(-1)

Value of absolute minimum = 0 

Question that needs to be answered is below

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.

To solve Question 4, let's analyze the situation based on the results from Question 3 and the function \( f(x, y) = e^{-(x^2 + y^2)}(x^2 + 2y^2) \).

### Step 1: Domain of the Function
The function is defined on the domain \( x^2 + y^2 \leq 4 \), meaning it's constrained within the circle of radius 2 centered at the origin.

### Step 2: Critical Points and Values
The given critical points are:
- \((-1, 0)\)
- \((0, -1)\)
- \((0, 0)\)
- \((0, 1)\)
- \((1, 0)\)

From Question 3, we know:
- The **absolute maximum** value is \( 2e^{-1} \), which occurs at one of the critical points.
- The **absolute minimum** value is 0, which occurs at the point \((0, 0)\).

### Step 3: Analyze Whether Values Occur on the Boundary
The boundary of the domain is defined by the circle \( x^2 + y^2 = 4 \), which corresponds to points at a distance of 2 from the origin.

- **The absolute minimum** of 0 occurs at \((0, 0)\), which is at the origin and **inside** the boundary, not on it.
- The **absolute maximum** \( 2e^{-1} \) occurs at one of the critical points, which are within the circle. However, none of these points lie on the boundary where \( x^2 + y^2 = 4 \).

### Step 4: Conclusion
Since neither the maximum nor the minimum values occur on the boundary, the correct answer is:

**a. Neither of the absolute max and min values of \( f \) occur on the boundary.**

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Would you like further details on any of these steps? 

Here are 5 related questions:
1. How do we find critical points of multivariable functions?
2. Why are some critical points local maxima or minima while others are saddle points?
3. How does the behavior of a function change along a boundary?
4. What is the significance of the boundary in constrained optimization problems?
5. How does the exponential decay factor \( e^{-(x^2 + y^2)} \) affect the function's values?

**Tip:** Always check the boundary when finding absolute extrema on a closed region, as maxima and minima might occur there.

Based on the information given solve the questions below

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

Q2) How many critical points does f have = 5 critical points, (−1,0), (0,−1), (0,0), (0,1) (1,0).

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 = 2exp(-1)

Value of absolute minimum = 0 

Question that needs to be answered is below

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.

Solve a multivariable calculus problem involving critical points and extrema of the function e^{-(x^2 + y^2)}(x^2 + 2y^2) within the domain x^2 + y^2 ≤ 4. Explore the absolute maximum of 2exp(-1) and minimum value of 0 with detailed analysis. Suitable for Calculus III students.

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