On-line Math 21

4.1  The Mean Value Theorem

The presentation here, as in the textbook, jumps into a different mode. The idea here is not just to describe, measure, and compute derivatives, but to use those ideas and derive consequences of them. This is a brief sojourn into theoretical mathematics.

is called a critical point of f(x) . The values of f(x) at a critical point are called critical values, or critical numbers.

Example 1 Find the local maximum and minimum points and values, and the absolute maximum minimum points and values, of the function

f(x) = 2sin(x)+cos(2x), x Î [0,2p].

Solution

This example illustrates a basic observation about critical points, usually called the first derivative test, since it tests whether a critical point is a local maximum or minimum by examining the first derivatives. From Corollary 3 of the href="mvt-corollaries.html" next section, the proof of this result is easy.

Theorem 2 [First Derivative Test]. If x₀ is a (non-endpoint) critical point of a function f(x) , and:

1) f¢(x) > 0 for x < x₀ (but x near x₀ ), and f¢(x) < 0 for x > x₀ , then x₀ is a local maximum point of f(x),

1) f¢(x) < 0 for x < x₀ (but x near x₀ ), and f¢(x) > 0 for x > x₀ , then x₀ is a local maximum point of f(x) .

I think of this as ``increasing up to a maximum, decreasing beyond it'', or ``decreasing down to a minimum, and increasing away''. It really is useful to find the signs of the derivative and lay them out on a number line as in the previous example,

There is another derivative test, called the second derivative test. It's not quite so good as this one, though, for two reasons. The first is that you have to assume that f¢(x₀) exists (note that the first derivative test does not mention that), and the second is that there is an indeterminate case.

Theorem 3 [Second Derivative Test]. If f¢(x₀) = 0, and

1) f¢¢(x₀) > 0 , then f(x) has a local minimum at x₀ ,

2) f¢¢(x₀) < 0 , then f(x) has a local maximum at x₀ .
But, if f¢¢(x₀) = 0 , it could be either a local maximum, or maybe a local minimum, or maybe neither one.

Exercise 5 Find the local and absolute maximum and minimum points, and their values, of the function

f(x) = x4-2x3+x2+1,

for x Î [-1,2] .

Answer:

Copyright (c) 2000 by David L. Johnson.