An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 21. 22.

Theorem of the Mean Value.

41

Note. This proof does not assume the continuity of f'(x), so
that f'(x) may become infinite in the interval. Moreover it does not
require f'(x) to have everywhere a determinate value; the only
assumption it makes is that
Lim f(x + h)-2 f(x) +f(x- ) _ O
hI0
at each point of the interval.
This Lemma furnishes the proof of the following proposition
which is called the Theorem of the Mean Value:
If f(x) be a unique function in the interval from a to b, whose
progressive and regressive differential quotients are everywhere in the
interval identical and determinate, then a value xl can always be found
between a and b such that we shall have the quotient of differences
f(b) - f(a)
b- a
For if we denote the value of the quotient of differences by K, so that:
{f(b)- b} - - {f(a) - Ka} 0,
and form the function:
q(x) = {f(b) - Kb } - {f(x) - Kx},
then this alike with f(x) will be continuous, it will likewise have
everywhere identical progressive and regressive differential quotients,
and furthermore it will have the same value 0 both for x a and
for x = b. Hence there must in the interval be a value x1 which
will make (p'(x) = 0.
But we have:
'(x,) ==K-f'(x,) =, that is: K- f(b)- =f'(x,) Q. E. D.)
*) Geometrically:
-Y J
x=ia x. b
Fig 4.
There is an intermediate point at which the tangent is parallel to the line joining
the extremities. Here also the case may occur that the derivate becomes infinite:
that is, that the tangent at a point is parallel to the axis of ordinates.
~*) This proof of the proposition, which is also called the theorem of Rolle
(1652-1719), is due to Serret (1819-85): Cours de calcul differentiel et integral,
2e ed., t. I, p. 17 seq.