Question

Use Lagrange multipliers to find the given extremum. In each case, assume that \(x\) and \(y\) are positive. $$ \text { Minimize } f(x, y)=x^{2}+y^{2} \quad x+y-4=0 $$

Short answer

The function \(f(x, y) = x^2 + y^2\) reaches its minimum value at the point (2, 2).

Step-by-step solution

Formulate the Lagrangian

First, form the Lagrangian function which is a function of the original variables and an additional variable, known as a Lagrange multiplier, denoted by \(\lambda\). The Lagrangian is defined as \(L(x, y, \lambda) = f(x, y) - \lambda * g(x, y)\). So, in this case, the Lagrangian is \(L(x, y, \lambda) = x^2 + y^2 - \lambda (x + y - 4)\).

Compute the gradients

Next, compute the gradient of the Lagrangian with respect to each of x, y and \(\lambda\), and then set each of these equal to zero to solve for x, y and \(\lambda\). This gives us the following system of equations: \(2x - \lambda = 0\); \(2y - \lambda = 0\); and \(x + y - 4 = 0\).

Solve the System of Equations

The first two equations give \(2x = \lambda\) and \(2y = \lambda\). This implies that \(2x = 2y\), so \(x = y\). Substituting x for y in the third equation gives \(2x - 4 = 0\), so \(x = y = 2\).

Key concepts

Extremum Optimization

Extremum optimization refers to the process of finding the maximum or minimum value of a function under given constraints. This is a common problem in disciplines such as economics, engineering, and physics, where it's essential to determine optimal conditions within certain limits.

The example provided in the textbook exercise involves finding the minimum value of the function f(x, y) = x^{2} + y^{2}, subject to the constraint x + y - 4 = 0. To solve this, we employ the method of Lagrange multipliers, which helps us to locate the minimum or maximum of f(x, y) while satisfying the constraint equation. It's a powerful technique because it lets us convert a constrained problem into an unconstrained one by introducing a new variable, the Lagrange multiplier.

Lagrangian Function

The Lagrangian function is a fundamental concept in optimization problems involving constraints. Formally, it combines the objective function and the constraint into a single expression by introducing a new variable, known as the Lagrange multiplier.

In the exercise, the Lagrangian function L(x, y, \(lambda\)) = f(x, y) - \(lambda\) * g(x, y) is built by subtracting the product of a Lagrange multiplier (here \(lambda\)) and the constraint function g(x, y) from the original function f(x, y). The purpose of this is to effectively 'ignore' the constraint by finding points where the gradient of the original function is parallel to the gradient of the constraint, indicating potential extremum points under the given constraint.

Gradients Computation

In the context of Lagrange multipliers, gradients computation involves taking the derivatives of the Lagrangian function with respect to each variable, including the Lagrange multipliers. The gradient is a vector that points in the direction of the greatest increase of the function, and its computation is crucial for finding the stationary points.

In the solution, computing the gradient with respect to x, y, and \(lambda\) and setting them to zero results in a system of equations. These equations are the necessary conditions for an extremum to exist under the given constraint. The gradient's components 2x - \(lambda\) = 0 and 2y - \(lambda\) = 0 show the relationship between the variables and the Lagrange multiplier at the extremum point for this particular problem.

Systems of Equations

The final stage in the Lagrange multipliers method involves solving a system of equations derived from the gradient computation. These equations reflect the stationary points where the Lagrangian function does not change, suggesting the presence of an extremum.

In our exercise, the systems of equations are 2x - \(lambda\) = 0, 2y - \(lambda\) = 0, and the constraint x + y - 4 = 0. Solving this system leads us to the values x = y = 2. This indicates that at the point (2, 2), the original function f(x, y) may have an extremum, while satisfying the given constraint. Such a system is key to the solution of constrained optimization problems, as it quantitatively connects the variables and the constraints.