Optimization in mathematical modeling refers to the process of identifying the most effective solution to a given problem.

In this context, optimization entails determining the maximum or minimum value of a function while adhering to specific constraints. This is typically achieved through various techniques, including differentiation, integration, and linear programming. The function that is being optimized is known as the objective function, while the constraints represent the limitations that must be fulfilled.

Consider the example of maximizing the area of a rectangle with a fixed perimeter of $20$ units. Let $x$ denote the length of the rectangle and $y$ represent its width. The perimeter can be expressed by the equation:

which simplifies to:

The area $A$ of the rectangle is given by the formula:

Our goal is to maximize $A$ under the constraint $x + y = 10$.

To express $y$ in terms of $x$, we can rearrange the constraint:

Substituting this expression into the area formula, we obtain:

To find the maximum area, we differentiate $A$ with respect to $x$ and set the derivative equal to zero:

Solving for $x$, we find:

Substituting this value back into the constraint equation yields:

Consequently, the maximum area of the rectangle is:

In summary, optimization in mathematical modeling involves finding the optimal solution to a problem by maximizing or minimizing a function while satisfying certain constraints. This process employs various mathematical techniques, such as differentiation and linear programming, to achieve the desired results.