2.10 The Mean Value Theorem

Summary

We can learn much about a function by studying its derivative. For example we can determine where the function is increasing, and where it is decreasing. And we can discover where it assumes its maximum and minimum values. The key to the relationship between such global properties of a function and the behavior of its derivative is the Mean Value Theorem. It will often arise in similar circumstances—when we need to connect local and global behavior of a function.

By the end of your studying, you should know:

A statement of the Mean Value Theorem (MVT) and how to interpret it.
When the hypotheses of the MVT hold, and when they don't.
The definitions of a function increasing on an interval, decreasing on an interval, nondecreasing on an interval, and nonincreasing on an interval.
How to use the derivative to test for an increasing, decreasing, nondecreasing, or noninceasing function on an interval.
How to find the intervals over which a function is increasing (or decreasing).
How to use the derivative to find the maximum or minimum value of a differentiable function on a closed and bounded interval.
Rolle's Theorem.

On-screen applet instructions: The tangent line, when you find it, will be solid and not dotted.

Examples

At 7 p.m., a car is traveling at 50 miles per hour. Ten minutes later, the car has slowed to 30 miles per hour. Show that at some time between 7 and 7:10 the car's acceleration is exactly 120, in units of miles per hours squared.

At a particular horse race, two horses start at the same time, and finish in a tie. Show that at some time during the race, the horses were running at the same speed.

Suppose f is a differentiable function such that

What is the smallest possible value for f(6)?