Objectives: In this tutorial, we investigate two important properties of functions which are continuous on a closed interval [a, b]: the Intermediate Value Theorem and the Extreme Value Theorem. As an application of the Intermediate Value Theorem, we discuss the existence of roots of continuous functions and the bisection method for finding roots.

After working through these materials, the student should be able to apply the Intermediate Value Theorem, to determine whether certain functions have roots, and use the bisection method to find roots to any degree of accuracy.

Modules:

 Intermediate Value Theorem. Let f be a function which is continuous on the closed interval [a, b]. Suppose that d is a real number between f(a) and f(b); then there exists c in [a, b] such that f(c) = d.

• Discussion [Using Flash]

• One of the useful consequences of the Intermediate Value Theorem is the following.

 Corollary. Let f be a function which is continuous on the closed interval [a, b]. Suppose that the product f(a) f(b) < 0; then there exists c in (a, b) such that f(c) = 0. In other words, f has at least one root in the interval (a, b)

• Example.

• Show that f(x) = x3 + 4x + 4 has a root.
[Solution]

• Using this Corollary, we can develop an algorithm for finding roots of functions to any degree of accuracy. This algorithm is called the Bisection Method.

Discussion [Using Flash]

• Interactive Javascript module illustrating the use of the Bisection Method.

 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 f is said to have a maximum value at x = a. 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 f is said to have a minimum value at x = a.

• Discussion [Using Flash]

• Another very important property of continuous functions defined on closed intervals is the following theorem.

 Extreme Value Theorem. Suppose that f is a function which is continuous on the closed interval [a, b]. Then there exist real numbers c and d in [a, b] such that f has a maximum value at x = c and f has a minimum value at x = d.

• While the Extreme Value Theorem may seem intuitively obvious, it is a difficult theorem to prove. The proof is usually covered in an advanced calculus or analysis class.

Techniques for finding c and d will be given after we develop more tools. Except for a few examples, we will rely now on graphical evidence for finding c and d.

• Examples:

• A quiz on using concepts involving continuity of functions and its consequences.