Math Problem Statement
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Solution
We need to find the minimum value of the function:
subject to the constraints:
To solve this, we can use the method of Lagrange multipliers. This involves introducing two Lagrange multipliers, and , for each constraint. Then, we set up the following system of equations:
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The gradients must satisfy:
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The constraints must hold:
Step 1: Calculate Gradients
The gradient of is:
The gradient of is:
The gradient of is:
Step 2: Set Up the System of Equations
Using the Lagrange multiplier equation , we get:
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For -component: which simplifies to:
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For -component: which simplifies to:
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For -component: which simplifies to:
Step 3: Solve the System with Constraints
We also have the constraint equations:
Now, we solve this system of equations. Let me work through the algebra.The solution to the system is:
Now, we can substitute these values into the function to find the minimum value.
Step 4: Calculate at the Solution Point
Let's substitute , , and into .The minimum value of the function subject to the given constraints is:
Would you like further details on any steps or have any additional questions?
Related Questions
- How does the method of Lagrange multipliers work in general?
- What other methods can be used to solve constrained optimization problems?
- How would the solution change if we modified the function ?
- Can we solve this problem using matrix methods?
- What would happen if one of the constraints was non-linear?
Tip
When using Lagrange multipliers, always check that the solution satisfies both the constraints and the original function's minimum requirements.
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Optimization
Lagrange Multipliers
Formulas
f(x, y, z) = x^2 + y^2 + z^2
∇f = λ∇g + μ∇h
Theorems
Lagrange Multiplier Method
Suitable Grade Level
Undergraduate Level
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