Calculus: Analytic Geometry and Calculus, with Vectors

5.5 The Rolle theorem and the mean-value theorem 329

(1) Theorem If f is such that f" exists over an interval containing a andx,
then there is at least one number x* between a and x such that

i a ii x*

f(x) = f(a) -I- (x - a) +f 2 !)(x - a) 2.

g(t) =f(x) - f(t) - j;) (x-t)-2 (x-t)2,

where C is a constant chosen such that ¢(a) = 0. Then ¢(a) = ¢(x) = 0 and
the other hypotheses of the Rolle theorem are satisfied. The Rolle theorem
therefore furnishes a number x between a and x for which t'(x) = 0. Thus,

(4) 0'(x*) _ -f'(x*) + f' (x*) - f"(x*)(x - x*) + C(x - x*) = 0.

Therefore, C = f"(x). Since ¢(a) = 0, we can put t = a in (3), equate the
result to 0, and transpose to obtain the required formula (2). Remark: With the
additional hypothesis that the second derivative f" is continuous, we shall use
integration by parts in Section 12.5 to obtain more straightforward derivations
of (2) and related formulas.
13 This problem requires attainment of understanding of matters relating to
the following generalized mean-value theorem which involves two functions.
(1) Theorem Let f and g be continuous over the closed interval from a to x,
let f and g be differentiable over the open interval from a to x, and let the derivative g'
be different from zero over the open interval from a to x. Then there exists an x
in the open interval from a to x such that

(2) f(x) - f(a)-_f'(x*)

g(x) - g(a)

g'(x*)

We assume that f and g are given functions satisfying the hypotheses of the

theorem. Two applications of the mean-value theorem then show that there

exist numbers xi and x2 between a and x such that

f(x) - f(a)

(3) f(x) - f(a)- x - a f'(xi),

g(x) - g(a) g(x) - g(a) g (x2)

x - a

While this result can be useful, it is crude because xi and x2 are not necessarily

equal. We obtain the more useful and elegant result (2) by arranging our work

to make a single application of the Rolle theorem. The trick is to define a new

function ¢ by the formula

(4) O(t) _ [f(x) - f(a)][g(t) - g(a)l - [f(t) - f(a)I [g(x) - g(a)1

It is easy to see that q 5(a) = O(x) = 0 and that the other hypotheses of the Rolle

theorem are satisfied. Hence the Rolle theorem implies that there is a number

x* between a and x for which

(5) 0'(t*) = U(x) - f(a)]g'(x*) - f'(x*)[g(x) - g(a)l = 0,