The Mean Value Theorem
Given a curve, y = f(x) we shall use the derivative to give us information about the curve. For instance, we shall find the maximum and minimum of the graph, and regions where the curve is increasing or decreasing. We shall use the mean value theorem, whi
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V
The Mean Value Theorem
Given a curve, y = f(x), we shall use the derivative to give us information about the curve. For instance, we shall find the maximum and minimum of the graph, and regions where the curve is increasing or decreasing. We shall use the mean value theorem, which is basic in the theory of derivatives.
V, §1. THE MAXIMUM AND MINIMUM THEOREM Definition. Let f be a differentiable function. A critical point of f is a number c such that f'(c) = O.
The derivative being zero means that the slope of the tangent line is 0 and thus that the tangent line itself is horizontal. We have drawn three examples of this phenomenon.
/~ I
\J I
(.
c
Figure 1
Figure 2
The third example is that of a function like f a (or x < a) an open interval. The context will always make this clear. Let f be a function, and c a number at which f is defined. Definition. We shall say that c is a maximum point of the function if and only if f(c)
~
f
f(x)
for all numbers x at which f is defined. If the condition f(c) ~ f(x) holds for all numbers x in some interval, then we say that the function has a maximum at c in that interval. We call f(c) a maximum value. Example 1. Let f(x) = sin x. Then f has a maximum at n/2 because f(n /2) = 1 and sin x ~ 1 for all values of x . This is illustrated in Fig. 4. Note that - 3n/2 is also a maximum for sin x .
",/ 2
Figure 4
Example 2. Let f(x) = 2x, and view f as a function defined only on the interval o ~ x ~ 2.
[V, §1]
THE MAXIMUM AND MINIMUM THEOREM
Then the function has a maximum at 2 in this interval because f(2) and f(x) ~ 4 for all x in the interval. This is illustrated in Fig. 5.
161
=4
4
2
Figure 5
Example 3. Let f(x) = l /x. We know that f is not defined for This function has no maximum. It becomes arbitrarily large when x comes close to 0 and x > O. This is illustrated in Fig. 6.
x
= O.
Figure 6
Definition. A minimum point for f(x)
~f(c)
f
is a number c such that
for all x where
f is defined.
A minimum value for the function is the value f(c), taken at a minimum point. We illustrate various minima with the graphs of certain functions.
Figure 7
Figure 8
In Fig. 7 the function has a minimum. In Fig. 8 the minimum is at the end point of the interval. In Figs. 3 and 6 the function has no minimum.
162
THE MEAN VALUE THEOREM
[V, §1]
In the following picture, the point C I looks like a maximum and the point C 2 looks like a minimum, provided we stay close to these points, and don't look at what happens to the curve farther away.
Figure 9 There is a name for such points. We shall say that a point c is a local minimum or relative minimum of the function f if there exists an interval
such that fCc) ~f(x) for all numbers x with a l ~ x ~ b l . Similarly, we define the notion of local maximum or relative maximum. (Do it yourself.) In Fig. 9, the point C3 is a local maximum, C4 is a local minimum, and C 5 is a local maximum. The actual maximum and minimum occur at the end points. Using basic properties of numbers, one can prove the next theorem which is
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