Math Problem Statement

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.

Solution

To solve Question 4, let's analyze the situation based on the results from Question 3 and the function f(x,y)=e−(x2+y2)(x2+2y2)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 x2+y2≤4x^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)(-1, 0)
  • (0,−1)(0, -1)
  • (0,0)(0, 0)
  • (0,1)(0, 1)
  • (1,0)(1, 0)

From Question 3, we know:

  • The absolute maximum value is 2e−12e^{-1}, which occurs at one of the critical points.
  • The absolute minimum value is 0, which occurs at the point (0,0)(0, 0).

Step 3: Analyze Whether Values Occur on the Boundary

The boundary of the domain is defined by the circle x2+y2=4x^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)(0, 0), which is at the origin and inside the boundary, not on it.
  • The absolute maximum 2e−12e^{-1} occurs at one of the critical points, which are within the circle. However, none of these points lie on the boundary where x2+y2=4x^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 ff occur on the boundary.

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−(x2+y2)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.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Critical Points
Optimization on a Closed Domain
Exponential Decay

Formulas

f(x, y) = e^{-(x^2 + y^2)}(x^2 + 2y^2)
x^2 + y^2 ≤ 4 (domain constraint)
e^x (exponential function)

Theorems

First Derivative Test for Critical Points
Extreme Value Theorem

Suitable Grade Level

University Level - Calculus III

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