Optimization Absolute Maxima and Minima

OPTIMIZATION ABSOLUTE MAXIMA AND MINIMA

Maximum and Minimum Values

A function f has an absolute maximum at c if f(c) ≥ f(x), for all x in the domain of f. At x=c, the graph reaches its highest point. The number f(c) is called the maximum value of f.

A function f has an absolute minimum at c if f(c) ≤ f(x), for all x in the domain of f. At x=c, the graph reaches its lowest point. The number f(c) is called the minimum value of f.

Together, the maximum and minimum values are called the extreme values of the function f.

GUIDELINES FOR SOLVING MAX./MIN. PROBLEMS

1. Read each problem slowly and carefully. Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.

2. If appropriate, draw a sketch or diagram of the problem to be solved. Pictures are a great help in organizing and sorting out your thoughts.

3. Define variables to be used and carefully label your picture or diagram with these variables. This step is very important because it leads directly or indirectly to the creation of mathematical equations.

4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize. Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint" equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations.

5. Before differentiating, make sure that the optimization equation is a function of only one variable. Then differentiate using the well-known rules of differentiation.

6. Verify that your result is a maximum or minimum value using the first or second derivative test for extreme values.