We have used derivatives to help find the maximums and minimums of some functions given by equations, but it is very unlikely that someone will simply hand you a function and ask you to find its extreme values. More typically, someone will describe a problem and ask your help in maximizing or minimizing something: What is the largest volume package which the post office will take? ; What is the quickest way to get from here to there? ; or What is the least expensive way to accomplish some task? In this section, we’ll discuss how to find these extreme values using calculus.
In business applications, we are often interested to maximize revenue, or maximize profit and minimize costs. For example, we can determine the derivative of the profit function and use this analysis to determine conditions to maximize profit levels for a business.
The manager of a garden store wants to build a 600 square foot rectangular enclosure on the store’s parking lot in order to display some equipment. Three sides of the enclosure will be built of redwood fencing, at a cost of $7 per running foot. The fourth side will be built of cement blocks, at a cost of $14 per running foot. Find the dimensions of the least costly such enclosure.
The process of finding maxima or minima is called optimization. The function we’re optimizing is called the objective function (or objective equation). The objective function can be recognized by its proximity to est words (greatest, least, highest, farthest, most, …). Look at the garden store example; the cost function is the objective function.
In many cases, there are two (or more) variables in the problem. In the garden store example again, the length and width of the enclosure are both unknown. If there is an equation that relates the variables we can solve for one of them in terms of the others, and write the objective function as a function of just one variable. Equations that relate the variables in this way are called constraint equations. The constraint equations are always equations, so they will have equals signs. For the garden store, the fixed area relates the length and width of the enclosure. This will give us our constraint equation.
