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Prereq: To understand this section, you should know how to locate relative and absolute maxima and minima of a real-valued function of a single variable. Press the "Prev Tutorial" button on the sidebar to go to the tutorial on that subject.
Applied problems in which we have to find the maximum or minimum are sometimes called optimization problems. The key to solving optimization problems is to set up the problems so that is amounts to locating the absolute maximum or minimum of a certain function f.
Sometimes, the given problem requires little or no setting up. here is one we can attempt right off the bat.
copies sold per week when the price is p dollars. What price should the company charge to obtain the largest revenue, and what is the largest revenue?
In questions of this kind, the function we are trying to optimize is called the objective function. Thus, the objective function if the function R. The endpoints and stationary points are given by:
Sometimes, the objective function may depend on more than one variable, as in the following applied optimization problem (taken from Example 2 in Section 5.2 of Applied Calculus).
The quantity A that we are trying to maximize is called the objective function. The conditions that come after "subject to" are called constraints. Thus, we are trying to maximize some objective function subject to one or more constraints:
Maximize A = xy subject to | Objective |
x + 2y = 100 | Equality constraint |
x ≤ 0 | Inequality constraint |
y ≤ 0 | Inequality constraint |
Q How do we solve optimization problems like this?
A Here's how:
The next example is similar to, but a little simpler than Example 5 in Section 5.2 of Applied Calculus or Section 13.2 of Finite Mathematics and Applied Calculus
You now have several options