Calculus: Analytic Geometry and Calculus, with Vectors

5.5 The Rolle theorem and the mean-value theorem 333

Theorem If f is integrable (Riemann) over a =< x 5 b and if F'(x)= f(x)

when a=< x<b, then

(1) ab f(x) dx= F(x)]a= F(b) - F(a).

Our proof of this theorem uses the mean-value theorem.Supposing that P

is a partition of the interval a 5 x < b having partition points xo, xi, x

for which

(2) a=xo<x,< ...

we find that

(3)

n+

F(b) - F(a) _ (F(xk) - F(xk-I)) = 1, F'(xk)(xk - xk-1)

k=1 k=1

when, for each k, xk is an appropriately chosen point in the interval xkI < x < xk.
Since F'(xk) = f(xk), the last sum is a Riemann sum formed for the function f
over the interval a < x 5 b. Since each sum is equal to F(b) - F(a), it follows
that the limit (which exists by hypothesis) of these sums must be F(b) - F(a).
The limit is the integral in (1) and the theorem is proved. Remark: The hypothe-
sis that F is differentiable and F'(x) = f(x) over a 5 x 5 b does not imply that
f is continuous over a 5 x < b; in fact the last of Problems 3.69 gives examples
of functions which are differentiable over the interval -1 <- x 5 1 but have
derivatives that are unbounded over this interval. Thus, some discontinuous
functions can be derivatives, but the following theorem shows that a function
cannot be a derivative unless it (like continuous functions) possesses the inter-
mediate-value property.
Theorem If F is differentiable over a < x 5 b and if F(a) < q < F(b) or
F'(a) > q > F'(b), then there is a number for which a < < b and q.
To prove this theorem, let g(x) = F(x) - q(x - a). Then g, like F, must be
continuous. Hence g(x) must attain a minimum value at some point for which
a < <- b. Consider the case in which F(a) < q < F'(b). Since g'(a)
F(a) - q < 0, we see that cannot be a. Since g'(b) = F(b) - q > 0, we see
that cannot be b. Hence a < t < b and therefore g'() = 0 and F'() = q.
In case F(a) > q > F'(b), g(x) must attain a maximum at a point for which
a < < b and F'(E) = q. This proves the theorem.
19 Prove that if f is continuous over - oo < x < oo, if f (x) -> 0 as x --+ oo,
and if f(x) -> 0 as x-> - oo, then f must have a global maximum or a global
minimum but not necessarily both. Hints: As in the proof of the Rolle theorem,
consider three cases. In case f(x) > 0 for at least one x, choose xo such that
f(xo) > 0. Choose a number H such that l f(x)I < If(xo) when jxj H.
The maximum of f(x) over the interval jxj 5 H must then be the maximum of
f(x) over the infinite interval.
20 Persons who manufacture peanut butter and typewriters and electronic
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of a commodity can be sold when the price is p(x) dollars per unit. The graph
of p versus x is the demand curve. The nature of the demand curve depends
upon the commodity, being relatively flat (or inelastic) for false teeth, since