Math Problem Statement

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Tentukan nilai minimum dari fungsi: f(x, y, z) = x^2 + y^2 + z^2 dengan kendala: g(x, y, z) = x + y + z - 8 dan h(x, y, z) = 2x - y + 3z - 28

Solution

We need to find the minimum value of the function:

f(x,y,z)=x2+y2+z2f(x, y, z) = x^2 + y^2 + z^2

subject to the constraints:

g(x,y,z)=x+y+z8=0g(x, y, z) = x + y + z - 8 = 0 h(x,y,z)=2xy+3z28=0h(x, y, z) = 2x - y + 3z - 28 = 0

To solve this, we can use the method of Lagrange multipliers. This involves introducing two Lagrange multipliers, λ\lambda and μ\mu, for each constraint. Then, we set up the following system of equations:

  1. The gradients must satisfy: f=λg+μh\nabla f = \lambda \nabla g + \mu \nabla h

  2. The constraints must hold: g(x,y,z)=0g(x, y, z) = 0 h(x,y,z)=0h(x, y, z) = 0

Step 1: Calculate Gradients

The gradient of ff is: f=(2x,2y,2z)\nabla f = (2x, 2y, 2z)

The gradient of gg is: g=(1,1,1)\nabla g = (1, 1, 1)

The gradient of hh is: h=(2,1,3)\nabla h = (2, -1, 3)

Step 2: Set Up the System of Equations

Using the Lagrange multiplier equation f=λg+μh\nabla f = \lambda \nabla g + \mu \nabla h, we get:

  1. For xx-component: 2x=λ1+μ22x = \lambda \cdot 1 + \mu \cdot 2 which simplifies to: 2x=λ+2μ2x = \lambda + 2\mu

  2. For yy-component: 2y=λ1+μ(1)2y = \lambda \cdot 1 + \mu \cdot (-1) which simplifies to: 2y=λμ2y = \lambda - \mu

  3. For zz-component: 2z=λ1+μ32z = \lambda \cdot 1 + \mu \cdot 3 which simplifies to: 2z=λ+3μ2z = \lambda + 3\mu

Step 3: Solve the System with Constraints

We also have the constraint equations:

x+y+z=8x + y + z = 8 2xy+3z=282x - y + 3z = 28

Now, we solve this system of equations. Let me work through the algebra.The solution to the system is:

x=4,y=2,z=6x = 4, \quad y = -2, \quad z = 6

Now, we can substitute these values into the function f(x,y,z)=x2+y2+z2f(x, y, z) = x^2 + y^2 + z^2 to find the minimum value.

Step 4: Calculate f(x,y,z)f(x, y, z) at the Solution Point

Let's substitute x=4x = 4, y=2y = -2, and z=6z = 6 into f(x,y,z)f(x, y, z).The minimum value of the function f(x,y,z)=x2+y2+z2f(x, y, z) = x^2 + y^2 + z^2 subject to the given constraints is:

f(4,2,6)=56f(4, -2, 6) = 56

Would you like further details on any steps or have any additional questions?

Related Questions

  1. How does the method of Lagrange multipliers work in general?
  2. What other methods can be used to solve constrained optimization problems?
  3. How would the solution change if we modified the function f(x,y,z)f(x, y, z)?
  4. Can we solve this problem using matrix methods?
  5. 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