Page 81 - Schaum's Outline of Theory and Problems of Advanced Calculus

72 DERIVATIVES [CHAP. 4

MEAN VALUE THEOREMS
These theorems are fundamental to the rigorous establishment of numerous theorems and formulas.
(See Fig. 4-5.)

y
B
C

E
f (b)
A
D
f (a)

x
a ξ b
Fig. 4-5

1. Rolle’s theorem.If f ðxÞ is continuous in ½a; bŠ and differentiable in ða; bÞ and if f ðaÞ¼ f ðbÞ¼ 0,
then there exists a point in ða; bÞ such that f ð Þ¼ 0.
0
Rolle’s theorem is employed in the proof of the mean value theorem. It then becomes a
special case of that theorem.
2. The mean value theorem. If f ðxÞ is continuous in ½a; bŠ and differentiable in ða; bÞ, then there
exists a point in ða; bÞ such that

f ðbÞ f ðaÞ
a < < b
0
b a ¼ f ð Þ ð16Þ
Rolle’s theorem is the special case of this where f ðaÞ¼ f ðbÞ¼ 0.
The result (16) can be written in various alternative forms; for example, if x and x 0 are in
ða; bÞ, then
between x 0 and x
0
f ðxÞ¼ f ðx 0 Þþ f ð Þðx x 0 Þ ð17Þ
We can also write (16)with b ¼ a þ h,in which case ¼ a þ h, where 0 < < 1.
The mean value theorem is also called the law of the mean.
3. Cauchy’s generalized mean value theorem. If f ðxÞ and gðxÞ are continuous in ½a; bŠ and differ-
entiable in ða; bÞ, then there exists a point in ða; bÞ such that
0
a < < b
f ðbÞ f ðaÞ f ð Þ
¼ ð18Þ
0
gðbÞ gðaÞ g ð Þ
where we assume gðaÞ 6¼ gðbÞ and f ðxÞ, g ðxÞ are not simultaneously zero. Note that the special
0
0
case gðxÞ¼ x yields (16).
L’HOSPITAL’S RULES
If lim f ðxÞ¼ A and lim gðxÞ¼ B, where A and B are either both zero or both infinite, lim f ðxÞ is
x!x 0 x!x 0 x!x 0 gðxÞ
often called an indeterminate of the form 0/0 or 1=1, respectively, although such terminology is
somewhat misleading since there is usually nothing indeterminate involved. The following theorems,
called L’Hospital’s rules, facilitate evaluation of such limits.
1. If f ðxÞ and gðxÞ are differentiable in the interval ða; bÞ except possibly at a point x 0 in this
interval, and if g ðxÞ 6¼ 0 for x 6¼ x 0 , then
0