Objectives: In this tutorial, we discuss the use of the derivative to detect whether a function is increasing or decreasing. In addition, we define the maximum, minimum, local maximum and local minimum of a function. After working through these materials, the student should be able

• to recognize graphically when a function is increasing or decreasing by looking at the graph of its derivative;
• to recognize graphically the local maximum and the local minimum of a function.

Modules:

• Recall the following definition.

 Definition. A function f is increasing on the interval I if, for each a < b in I, f(a) < f(b). A function f is decreasing on the interval I if, for each a < b in I, f(a) > f(b).

• Intuitively, by looking at the graphs of increasing and decreasing functions, the following theorem appears to be reasonable.

 Theorem. If the derivative of a function f is positive for all x in an interval I, then f is increasing on I. If the derivative of a function f is negative for all x in an interval I, then f is decreasing on I.

• The proof of this Theorem will be given in another module.

• Example.

• Using a graphing calculator to graph a function and its derivative.

 Definition. Suppose that a is in the domain of the function f such that, for all x in the domain of f, f(x) < f(a) then a is called a maximum of f. Suppose that a is in the domain of the function f such that, for all x in the domain of f, f(x) > f(a) then a is called a minimum of f.

• Finding maxima and minima is one of the major problems which calculus helps us solve.

 Definition. Suppose that a is in the domain of the function f and suppose that there is an open interval I containing a which is contained in the domain of f such that, for all x in I, f(x) < f(a) then a is called a local maximum of f. Suppose that a is in the domain of the function f and suppose that there is an open interval I containing a which is also contained in the domain of f such that, for all x in I, f(x) > f(a) then a is called a local minimum of f.

• Discussion [Using Flash]