Math Problem Statement

Use Lagrange multiplier to find the maximum and minimum values of the function f(x,y,z) subject to the given constraint. f(x,y,z)=xyz , x^2+y^2+z^2=12.

Solution

To find the maximum and minimum values of the function f(x,y,z)=xyzf(x, y, z) = xyz subject to the constraint x2+y2+z2=12x^2 + y^2 + z^2 = 12, we will use the method of Lagrange multipliers. The method involves introducing a Lagrange multiplier λ\lambda and solving the system of equations given by the gradients of the function and the constraint.

  1. Define the Lagrangian function: L(x,y,z,λ)=xyz+λ(12x2y2z2)\mathcal{L}(x, y, z, \lambda) = xyz + \lambda (12 - x^2 - y^2 - z^2)

  2. Compute the partial derivatives of L\mathcal{L} with respect to xx, yy, zz, and λ\lambda, and set them equal to zero: Lx=yz2λx=0\frac{\partial \mathcal{L}}{\partial x} = yz - 2\lambda x = 0 Ly=xz2λy=0\frac{\partial \mathcal{L}}{\partial y} = xz - 2\lambda y = 0 Lz=xy2λz=0\frac{\partial \mathcal{L}}{\partial z} = xy - 2\lambda z = 0 Lλ=12x2y2z2=0\frac{\partial \mathcal{L}}{\partial \lambda} = 12 - x^2 - y^2 - z^2 = 0

  3. Solve the system of equations:

From yz2λx=0yz - 2\lambda x = 0: yz=2λxyz = 2\lambda x

From xz2λy=0xz - 2\lambda y = 0: xz=2λyxz = 2\lambda y

From xy2λz=0xy - 2\lambda z = 0: xy=2λzxy = 2\lambda z

From the constraint: x2+y2+z2=12x^2 + y^2 + z^2 = 12

  1. Express λ\lambda in terms of x,y,zx, y, z:

Dividing the first two equations: yzxz=2λx2λyyx=xyy2=x2y=±x\frac{yz}{xz} = \frac{2\lambda x}{2\lambda y} \Rightarrow \frac{y}{x} = \frac{x}{y} \Rightarrow y^2 = x^2 \Rightarrow y = \pm x

Dividing the first and third equations: yzxy=2λx2λzzx=xzz2=x2z=±x\frac{yz}{xy} = \frac{2\lambda x}{2\lambda z} \Rightarrow \frac{z}{x} = \frac{x}{z} \Rightarrow z^2 = x^2 \Rightarrow z = \pm x

Considering y=xy = x and z=xz = x, we get: x2+x2+x2=123x2=12x2=4x=±2x^2 + x^2 + x^2 = 12 \Rightarrow 3x^2 = 12 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2

Thus, we have the points (2,2,2)(2, 2, 2) and (2,2,2)(-2, -2, -2).

Considering y=xy = -x and z=xz = x, we get: x2+(x)2+x2=123x2=12x2=4x=±2x^2 + (-x)^2 + x^2 = 12 \Rightarrow 3x^2 = 12 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2

Thus, we have the points (2,2,2)(2, -2, 2) and (2,2,2)(-2, 2, -2).

Considering y=xy = x and z=xz = -x, we get: x2+x2+(x)2=123x2=12x2=4x=±2x^2 + x^2 + (-x)^2 = 12 \Rightarrow 3x^2 = 12 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2

Thus, we have the points (2,2,2)(2, 2, -2) and (2,2,2)(-2, -2, 2).

  1. Evaluate f(x,y,z)=xyzf(x, y, z) = xyz at these points:

For (2,2,2)(2, 2, 2): f(2,2,2)=222=8f(2, 2, 2) = 2 \cdot 2 \cdot 2 = 8

For (2,2,2)(-2, -2, -2): f(2,2,2)=(2)(2)(2)=8f(-2, -2, -2) = (-2) \cdot (-2) \cdot (-2) = -8

For (2,2,2)(2, -2, 2): f(2,2,2)=2(2)2=8f(2, -2, 2) = 2 \cdot (-2) \cdot 2 = -8

For (2,2,2)(-2, 2, -2): f(2,2,2)=(2)2(2)=8f(-2, 2, -2) = (-2) \cdot 2 \cdot (-2) = 8

For (2,2,2)(2, 2, -2): f(2,2,2)=22(2)=8f(2, 2, -2) = 2 \cdot 2 \cdot (-2) = -8

For (2,2,2)(-2, -2, 2): f(2,2,2)=(2)(2)2=8f(-2, -2, 2) = (-2) \cdot (-2) \cdot 2 = 8

  1. Conclusion:

The maximum value of f(x,y,z)=xyzf(x, y, z) = xyz is 88, and the minimum value is 8-8.

Do you want more details or have any questions?

Here are some other problems you can ask about:

  1. How to solve a system of linear equations using matrices?
  2. Explain the concept of eigenvalues and eigenvectors.
  3. What is the method of integration by parts?
  4. How to find the roots of a polynomial equation?
  5. Explain the concept of probability density function.

Tip: When working with constraints in optimization problems, always check the boundary conditions if they are within the domain of the problem.

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Math Problem Analysis

Mathematical Concepts

Lagrange multipliers
Optimization
Partial derivatives
Constraint equations

Formulas

Lagrangian function
Gradient equations

Theorems

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Suitable Grade Level

Advanced Undergraduate