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Part 1
Find the extremum of f(x,y) subject to the given constraint, and state whether it is a maximum or a minimum.
f left parenthesis x comma y right parenthesis equals 2 x squared plus 4 y squaredf(x,y)=2x2+4y2;
4 x plus 4 y equals 2884x+4y=288
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Part 1
There is a
▼
value of
enter your response here
located at (x,
y)equals=enter your response here.
(Simplify your answers.)
Solution
To solve the given optimization problem using the method of Lagrange multipliers, we proceed as follows:
Problem Statement:
We are tasked to find the extremum of:
subject to the constraint:
Step 1: Lagrange Multiplier Method
Define the Lagrange function:
where is the Lagrange multiplier.
Step 2: Partial Derivatives
Compute the partial derivatives of with respect to , , and :
Step 3: Solve the System of Equations
From and , we equate:
Substitute into the constraint :
Substitute back into :
Step 4: Classify the Extremum
Substitute and into :
Thus, the extremum value of is .
Step 5: Nature of the Extremum
The function is a sum of squares, which is always non-negative. Hence, it has a global minimum at .
Final Answer:
- Value of the extremum:
- Location:
- Nature: Minimum
Would you like a more detailed explanation or visualization of the solution?
Related Questions:
- What is the purpose of the Lagrange multiplier method in optimization problems?
- How do you interpret the constraint in real-world scenarios?
- Can you explain why reaches a minimum and not a maximum in this case?
- What happens to the solution if the constraint changes to for another ?
- How would this problem change if were not a quadratic function?
Tip:
Always simplify constraints early in optimization problems to reduce algebraic complexity.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Optimization
Lagrange Multipliers
Formulas
f(x, y) = 2x^2 + 4y^2
Constraint: x + y = 72
Lagrange Function: 𝓛(x, y, λ) = f(x, y) + λ(constraint)
Theorems
Lagrange Multipliers Theorem
Suitable Grade Level
Undergraduate Calculus
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