The Lagrange multiplier is a mathematical technique used in optimization problems to find the maximum or minimum of a function subject to constraints. It introduces an auxiliary variable, called the multiplier, which helps incorporate the constraints into the optimization process. This method is particularly useful when dealing with functions of multiple variables. In essence, the Lagrange multiplier allows us to transform a constrained problem into an unconstrained one by creating a new function, known as the Lagrangian. By setting the gradient of the Lagrangian to zero, we can find the optimal points while satisfying the given constraints.