Lagrange multipliers are a mathematical tool used to find the maximum or minimum value of a function subject to constraints. Essentially, they allow us to find the optimal solution to a problem by combining multiple equations and constraints.

For example, let’s say we have a rectangular field that we want to fence off. We have a limited amount of fencing material and want to find the dimensions of the field that will give us the maximum area. We can use Lagrange multipliers to solve this problem.

First, we set up our objective function, which is the area of the rectangular field. This is given by the formula A = lw, where l is the length and w is the width of the field.

Next, we set up our constraint equation, which is the amount of fencing material we have available. Let’s say we have 100 feet of fencing material. This means that the perimeter of the field must be equal to 100 feet, or 2l + 2w = 100.

Now, we can use Lagrange multipliers to find the optimal dimensions of the field that will give us the maximum area. We do this by combining our objective function and constraint equation into a single equation: A – λ(2l + 2w – 100) = 0.

Solving this equation, we find that the optimal dimensions of the field are l = 25 and w = 20, which gives us an area of 500 square feet.

Another example of using Lagrange multipliers is in finding the shortest distance between two points. Let’s say we want to find the shortest distance between the points (2,3) and (5,7). We can use Lagrange multipliers to solve this problem.

First, we set up our objective function, which is the distance between the two points. This is given by the formula d = √((x1-x2)^2+(y1-y2)^2), where x1 and y1 are the coordinates of the first point, and x2 and y2 are the coordinates of the second point.

Next, we set up our constraint equation, which is the fact that the two points are fixed. This means that x1 and y1 are fixed at the coordinates (2,3), and x2 and y2 are fixed at the coordinates (5,7).

Now, we can use Lagrange multipliers to find the shortest distance between the two points. We do this by combining our objective function and constraint equation into a single equation: d – λ(x1 – 2) – λ(y1 – 3) – λ(x2 – 5) – λ(y2 – 7) = 0.

Solving this equation, we find that the shortest distance between the two points is d = √(3^2 + 4^2) = 5.

In both of these examples, Lagrange multipliers allowed us to find the optimal solution to the problem by combining multiple equations and constraints. In the first example, we found the dimensions of a rectangular field that would give us the maximum area, and in the second example, we found the shortest distance between two points. Lagrange multipliers are a powerful tool that can help us find the optimal solutions to a wide range of problems.