324 Functions, graphs, and numbersconstant c for which f(x) = c when a < x < b? Does the hypothesis that
Gf"(x) = 0 when a < x < b imply that there are constants cl and c2 for which
f(x) = cix + C2 when a < x < b?
5.5 The Rolle theorem and the mean-value theorem In this
section we prove some fundamental theorems and use them to review and
prove some theorems that have been previously given without proof.
The following theorem must be permanently remembered and known as
the Rolle (Michel, 1652-1719) theorem. It is not to be presumed that
Rolle proved or even knew this theorem, but he did discover some of its
applications to polynomials.
Theorem 5.51 (Rolle theorem) If a < b, if f is continuous over
a =< x <_ b, if f is differentiable over a < x < b, and if f(a) = f(b) = 0,
then there is at least one number x* for which a < x* < b and f'(x*) = 0.
The proof of this theorem is mildly tricky because it seems to be
necessary to consider three different cases. Suppose first that f(x) = 0
over the whole interval a < x <_ b. Then f'(x) = 0 when a < x < b
and we can choose x* to be any number between a and b. Suppose next
that there is a number x1 for which a < xl < b and f(xl) > 0. Then
with the aid of Theorem 5.47 we see that f must attain a positive maxi-
mum f(x*) at some point x* for which a < x* < b, and we can be sure
that a < x* < b because f(a) = f(b) = 0. Since f' (x*) exists, it follows
from Theorem 5.26 that f' (x*) = 0. Suppose finally that there is a
number x2 for which a< x2 < b and f(x2) < 0. Arguments similar to
those used above then show that f must have a negative minimum at
some point x* for which a < x* < b and that f'(x*) = 0.
The following theorem is known as the law of the mean or the mean-
value theorem of the derivative calculus. It is a strengthened version of
Theorem 5.41, which we have discussed briefly, the right member of
(5.53) being the slope of the chord joining the points (a, f(a)) and (b,
f(b)). It, like the Rolle theorem, must be permanently remembered.
Theorem 5.52 (mean-value theorem) If a < b, if f is continuous
over a < x < b, and if f is differentiable over a < x < b, then there is at
least one number x* for which a < x* < b and-
(5.53) f(a)
fF(x*) =f (bb
a
or(5.54) f (b) - f (a) = f' (x*) (b - a).
This theorem differs from the Rolle theorem because it is not assumed
that f(a) = f(b) = 0. It happens, however, that the theorem can be
proved by applying the Rolle theorem to the function 0 defined by
O(x) = f(x) - g(x), where g is the function whose graph is the chord