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## Optimization

An important class of engineering problems involves the concept of optimization. In optimization problems you are given a process or set of processes that have offsetting features so that the best solution is some specific combination of variables that describe the process. Entire courses are taught about the concepts of optimization. For our purposes we will consider the simplest case where a process can be described by an expression with a single variable.

The steps necessary to solve optimization problems are:

1. Define a mathematical expression that governs the process that you are studying. This is called modeling.
2. Plot the expression to see if and where it has minima and maxima.
3. Take the derivative of the expression with respect to the independent variable.
4. Form an equation by setting the expression for the derivative equal to zero.
5. Solve for the roots of the equation formed in previous step. Recall from calculus that the derivative of an expression equals zero where it has a relative or absolute minimum or maximum because the slope is zero at those points and slope is the derivative.
6. Examine the roots found in previous step. Evaluate the original expression at any root(s) of interest to find the maximum or minimum value of the expression at that root.

Notice, there are no new commands to learn. The steps necessary to solve an optimization problem use the Maple commands that you have learned in previous lessons. The important part is remembering each step and the order to perform them to obtain a minima or maxima. This is also known as an algorithm. Algorithms will be discussed in more detail in a later module.