# Lagrange multiplier

A **Lagrange multiplier** is a constant, usually represented by the Greek symbol lambda (λ), which is part of the formula for finding an extreme value of a function *f(x,y)* subject to a constraint *g(x,y)*. For example, the method of Lagrange multipliers can help determine the maximum area of a rectangle that can be fit inside a given circle represented by *g(x,y) = x ^{2} + y^{2}*.

The points at which extrema occur for a function *f(x,y)* subject to the constraint *g(x,y)* are the common solutions to:

and *g(x,y) = 0*.

## Example

*Problem*: What is the shape and dimension of the largest rectangle that fits within a circle having a radius equal to the square root of 2?

*Solution*: *g(x,y) = x ^{2} + y^{2} - 2 = 0*. We want to maximize the area of a rectangle. Assuming symmetry about the origin (we could always map a solution to a symmetric one), the area

*f(x,y)*that we want to maximize equals 2x times 2y:

*f(x,y) = 4xy*. From the equation for the Lagrange multiplier above, and setting the

*x*and

*y*coefficients equal to each other, we can generate these two equations:

- 4y = λ2x
- 4x = λ2y

Solving each for lambda (λ) demonstrates that x must be equal to y, and plugging that into the equation for *g* yields:

- 2x
^{2}= 2 - x = 1 = y

Then do not forget to include the maximum area in your answer, which is .