The function f(x,y) = 6xy has an absolute maximum value and absolute minimum value subject to the constraint 3x² +3y² - 5xy=121. Use Lagrange multipliers to find these values. Find the gradient of f(x,y)=6xy. Vf(x,y) = (6y,6x) Find the gradient of g(x,y)=3x² +3y² -5xy-121. Vg(x,y) = (6x-5y,6y-5x) Write the Lagrange multiplier conditions. Choose the correct answer below. A. 6y=2(6x-5y), 6x = 2(6y-5x), 3x² + 3y² -5xy - 121=0 O B. 6x=2(6x-5y), 6y=2(6y-5x), 3x² + 3y²-5xy-121=0 OC. 6xy=2(6x-5y), 6xy=2(6y-5x), 3x² + 3y² -5xy-121=0 O D. 6x=2(6x-5y), 6y=2(6y-5x), 6xy=0 The absolute maximum value is (…)
The function f(x,y) = 6xy has an absolute maximum value and absolute minimum value subject to the constraint 3x² +3y² - 5xy=121. Use Lagrange multipliers to find these values. Find the gradient of f(x,y)=6xy. Vf(x,y) = (6y,6x) Find the gradient of g(x,y)=3x² +3y² -5xy-121. Vg(x,y) = (6x-5y,6y-5x) Write the Lagrange multiplier conditions. Choose the correct answer below. A. 6y=2(6x-5y), 6x = 2(6y-5x), 3x² + 3y² -5xy - 121=0 O B. 6x=2(6x-5y), 6y=2(6y-5x), 3x² + 3y²-5xy-121=0 OC. 6xy=2(6x-5y), 6xy=2(6y-5x), 3x² + 3y² -5xy-121=0 O D. 6x=2(6x-5y), 6y=2(6y-5x), 6xy=0 The absolute maximum value is (…)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Optimizing a Function Using Lagrange Multipliers**
The function \( f(x, y) = 6xy \) has an absolute maximum value and an absolute minimum value subject to the constraint \( 3x^2 + 3y^2 - 5xy = 121 \). Use Lagrange multipliers to find these values.
**Steps:**
1. **Find the gradient of \( f(x, y) = 6xy \):**
\[
\nabla f(x, y) = \langle 6y, 6x \rangle
\]
2. **Find the gradient of \( g(x, y) = 3x^2 + 3y^2 - 5xy - 121 \):**
\[
\nabla g(x, y) = \langle 6x - 5y, 6y - 5x \rangle
\]
3. **Write the Lagrange multiplier conditions. Choose the correct answer below:**
A. \(\ 6y = \lambda(6x - 5y), \ 6x = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
B. \(\ 6x = \lambda(6x - 5y), \ 6y = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
C. \(\ 6xy = \lambda(6x - 5y), \ 6xy = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
D. \(\ 6x = \lambda(6x - 5y), \ 6y = \lambda(6y - 5x), \ 6xy = 0\)
- **Correct Answer: A**
4. **Result:**
The absolute maximum value is \(\square\). (This space is left for users to compute the numerical value based on algebraic manipulation.)
This educational page guides users through finding extreme values of a function under a constraint using the method of Lagrange multip](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F511d779c-bf10-4315-98b1-4f0038c55113%2Ff1eaec0f-d9b4-4911-a0fc-a9d2543c5070%2F9bdbc5o_processed.png&w=3840&q=75)
Transcribed Image Text:**Optimizing a Function Using Lagrange Multipliers**
The function \( f(x, y) = 6xy \) has an absolute maximum value and an absolute minimum value subject to the constraint \( 3x^2 + 3y^2 - 5xy = 121 \). Use Lagrange multipliers to find these values.
**Steps:**
1. **Find the gradient of \( f(x, y) = 6xy \):**
\[
\nabla f(x, y) = \langle 6y, 6x \rangle
\]
2. **Find the gradient of \( g(x, y) = 3x^2 + 3y^2 - 5xy - 121 \):**
\[
\nabla g(x, y) = \langle 6x - 5y, 6y - 5x \rangle
\]
3. **Write the Lagrange multiplier conditions. Choose the correct answer below:**
A. \(\ 6y = \lambda(6x - 5y), \ 6x = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
B. \(\ 6x = \lambda(6x - 5y), \ 6y = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
C. \(\ 6xy = \lambda(6x - 5y), \ 6xy = \lambda(6y - 5x), \ 3x^2 + 3y^2 - 5xy - 121 = 0\)
D. \(\ 6x = \lambda(6x - 5y), \ 6y = \lambda(6y - 5x), \ 6xy = 0\)
- **Correct Answer: A**
4. **Result:**
The absolute maximum value is \(\square\). (This space is left for users to compute the numerical value based on algebraic manipulation.)
This educational page guides users through finding extreme values of a function under a constraint using the method of Lagrange multip
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