Problem 4 - A First Course in Mathematical Modeling

Question

Use the method of Lagrange multipliers to solve Problems. Find three numbers whose sum is 9 and whose sum of squares is as small as possible.

Step-by-Step Solution

Answer

The three numbers are 3, 3, and 3.

Step 1: Define Variables and Objective Function

Let the three numbers be denoted as \( x, y, \) and \( z \). Our objective is to minimize the sum of the squares of these numbers, which can be expressed as the function \( f(x, y, z) = x^2 + y^2 + z^2 \).

Step 2: State the Constraint

We are given that the sum of the three numbers is 9. Therefore, the constraint can be written as \( g(x, y, z) = x + y + z - 9 = 0 \).

Step 3: Set up the Lagrangian

The Lagrangian is \( \mathcal{L}(x, y, z, \lambda) = x^2 + y^2 + z^2 + \lambda(x + y + z - 9) \), where \( \lambda \) is the Lagrange multiplier.

Step 4: Find the Partial Derivatives

Compute the partial derivative of the Lagrangian with respect to each variable and equate them to zero: \( \frac{\partial \mathcal{L}}{\partial x} = 2x + \lambda = 0 \) \( \frac{\partial \mathcal{L}}{\partial y} = 2y + \lambda = 0 \) \( \frac{\partial \mathcal{L}}{\partial z} = 2z + \lambda = 0 \) \( \frac{\partial \mathcal{L}}{\partial \lambda} = x + y + z - 9 = 0 \)

Step 5: Solve the System of Equations

From \( 2x + \lambda = 0 \), \( 2y + \lambda = 0 \), \( 2z + \lambda = 0 \), we get \( x = y = z \). Substitute into the constraint: \( 3x = 9 \) So, \( x = y = z = 3 \).

Step 6: Verify Solution

Substitute \( x = 3, y = 3, z = 3 \) back into the constraint to verify: \( 3 + 3 + 3 = 9 \). The constraint is satisfied, confirming the solution is correct.

Key Concepts

Optimization

Optimization is the process of making something as effective or functional as possible. In mathematics, it often involves finding the maximum or minimum value of a function. In our problem, we are looking to minimize the sum of squares of three numbers. This means finding the smallest possible value of the function \( f(x, y, z) = x^2 + y^2 + z^2 \). This is a common type of problem in optimization, where you might also seek to maximize a certain result.

• Objective Function: The function you want to optimize. Here, it's \( x^2 + y^2 + z^2 \).
• Constraints: The conditions that must be met while optimizing. For us, it's that the sum of the numbers equals 9.

Understanding which quantities to optimize and under what conditions gives structure to solving optimization problems. This is the cornerstone of many mathematical and real-world applications, from minimizing cost to maximizing efficiency.

Constrained Optimization

Constrained optimization involves optimizing a function subject to one or more constraints. Constraints are conditions that your solution must satisfy. In our exercise, the constraint is that the sum of three numbers must be 9. A powerful method to solve such problems is using Lagrange multipliers.
Lagrange multipliers work by introducing a new variable \( \lambda \), the Lagrange multiplier. This accounts for how the constraints influence the optimization. Here's how it works:

• Lagrangian Function: \( \mathcal{L}(x, y, z, \lambda) = x^2 + y^2 + z^2 + \lambda(x + y + z - 9) \).
• Equations: You'll set up partial derivatives with respect to each variable and equate them to zero.

By solving these equations, you find the points that satisfy both the optimization goal and the constraints. In our problem, it showed that all three numbers should be equal for minimal sum of squares. This technique is a staple in multivariable calculus when constraints are involved.

Multivariable Calculus

Multivariable calculus is an extension of calculus dealing with functions of multiple variables. It provides the tools required for exploring more complex systems where multiple influencing factors are at play. In our problem, we have three variables \( x, y, \) and \( z \), and we need to consider how changes in these variables affect the objective function.
Key components include:

• Partial Derivatives: These allow us to see how the function changes with respect to one variable at a time, keeping others constant.
• Gradient: A vector composed of the partial derivatives, revealing the direction of maximum increase of the function.

Using tools from multivariable calculus, such as setting partial derivatives to zero, allows us to identify critical points where the function might achieve its extremum (maximum or minimum). In conjunction with constraints, as managed through Lagrange multipliers, these concepts enable us to solve complex optimization problems efficiently, even when multiple factors are involved.