Question

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x, y\), and \(z\) are positive. Maximize \(f(x, y, z)=x^{2} y^{2} z^{2}\) Constraint: \(x^{2}+y^{2}+z^{2}=1\)

Short answer

The maximum value of the function \(f(x, y, z)=x^{2} y^{2} z^{2}\) subject to the constraint \(x^{2}+y^{2}+z^{2}=1\) is 1/27 and occurs at x = y = z = 1/√3

Step-by-step solution

State the function and the constraint

Here, the function to be maximized is \(f(x, y, z)=x^{2} y^{2} z^{2}\) and the constraint is \(g(x, y, z) = x^{2}+y^{2}+z^{2}=1\)

Set up the Lagrange function

The Lagrange function is \(L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 1)\), where λ is the Lagrange multiplier. So, the Lagrange function here is \(L(x, y, z, λ) = x^{2} y^{2} z^{2} - λ(x^{2} + y^{2} + z^{2} - 1)\)

Compute the partial derivatives

Find the partial derivatives of L with respect to a given variable and set each of them equal to zero. These are: \( ∂L/∂x = 2x*y^2*z^2 - 2λx = 0\), \( ∂L/∂y = 2y*x^2*z^2 - 2λy = 0\), \( ∂L/∂z = 2z*x^2*y^2 - 2λz = 0\) and \( ∂L/∂λ = 1 - x^2 - y^2 - z^2 = 0\)

Solve the system of equations

As a result of Step 3, four equations are obtained. From the first three equations, one can find that \(λ = x^2y^2z^2\). Plugging this into the fourth equation gives \(x^4y^2z^2 + x^2y^4z^2 + x^2y^2z^4 = 1\). Solving this single equation of multiple variables requires certain assumptions which are provided in the problem. In this case, the solution is \(x = y = z = 1/√3\)

Substitute back into the function

Upon obtaining \(x\), \(y\), and \(z\), substitute these values back into the function to find the maximum value of \(f\). In this case, \(f(1/√3, 1/√3, 1/√3) = ((1/√3)^2*(1/√3)^2*(1/√3)^2) = 1/27\)

Key concepts

Constraint Optimization

Constraint Optimization is a type of problem where you are trying to find the maximum or minimum value of a function given certain limitations or constraints. In the context of multivariable calculus, this involves functions of several variables. In our exercise, we aim to maximize the function \(f(x, y, z) = x^2 y^2 z^2\) subject to the constraint \(x^2 + y^2 + z^2 = 1\). This constraint represents a sphere in three-dimensional space.
The method of Lagrange multipliers is particularly useful in solving such problems where the constraint is an equation. Instead of directly handling the constraint, we incorporate it into a new function, the Lagrange function, which allows us to use calculus to find the optimum points. Essentially, Lagrange multipliers assess how much the constrained function would change with a small increase in the constraint.
Applying Lagrange multipliers entails setting up a system of equations derived from the function and its constraint. Solving these equations helps find the points that satisfy both the function's maximum or minimum values and the constraint simultaneously.

Multivariable Calculus

Multivariable Calculus is the extension of calculus to functions of several variables. It encompasses a variety of operations such as partial differentiation, which is crucial for problems involving functions with more than one variable, like the one in this exercise. Here, the challenge is to optimize the function \(f(x, y, z) = x^2 y^2 z^2\) where \(x\), \(y\), and \(z\) are variables subject to a constraint.
In this scenario, each variable contributes to the overall behavior of the function, making it necessary to consider their interactions. Calculating partial derivatives allows us to observe how changes in one variable affect the function when the others are held constant.

• Partial derivatives help in building the Lagrange function, which incorporates both the objective function and the constraint.
• This is key in determining stationary points where potential extrema may occur.

Understanding multidimensional interactions of functions is crucial for tackling more complex calculus problems, particularly in the fields of engineering and economics.

Extremum Problems

Extremum problems focus on finding maximum or minimum values of a function. In our example, we are trying to find the maximum of a function given a constraint. Such problems are common in optimization tasks and are crucial for decision-making strategies in various domains.
For this exercise, the goal is to find the maximum value of the production function \(f(x, y, z) = x^2 y^2 z^2\) under the constraint \(x^2 + y^2 + z^2 = 1\). The presence of the constraint necessitates using the method of Lagrange multipliers.

• The first step is finding partial derivatives to construct the Lagrange function.
• Next, solving the resulting system of equations gives potential points where extrema can occur.

In such tasks, understanding the geometric interpretation is equally important, as these problems often represent practical constraints in real-world scenarios, such as limited resources or environmental conditions. Recognizing these aspects can make finding solutions a more intuitive process.