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(ii) Supposethat
x
y
, and let
s
x
)
y
.
Show that the inequality in (i) is equivalent to
(1
f
(
s
)
f
(
x
)
s
x
f
(
y
)
f
(
s
)
y
s
.
(iii) If thefunction
f
is differentiable on an interval including
x
and
y
, use the mean value theorem to show that the
inequality in (ii) would follow if it was known that
f
was
strictly increasing.
(iv) If
f
(
x
)
0 on an interval, use qn 15 to deduce that
f
is
concaveupwards on that intrval.
Cauchy's Mean Value Theorem
25 (
Cauchy
, 1823) Functions
f
:[
a
,
b
]
R and
g
:[
a
,
b
]
R are
continuous on the closed interval [
a
,
b
] and differentiable on the
open interval (
a
,
b
) and
g
(
x
)
0 for any
x
[
a
,
b
]. Let
f
(
b
)
f
(
a
)
g
(
b
)
g
(
a
)
K
.
Prove that the conditions of Rolle's Theorem hold for the function
F
:[
a
,
b
]
R defined by
F
(
x
)
f
(
b
)
f
(
x
)
K
(
g
(
b
)
g
(
x
)).
Apply Rolle's Theorem to
F
to show that, for some
c
such that
a
c
b
,
(
c
)
g
(
c
)
.
f
K
de l'Hoˆ pital's rule
26 Functions
f
:[
a
,
b
]
R and
g
:[
a
,
b
]
R arecontinuous on the
closed interval [
a
,
b
] and differentiable on the open interval (
a
,
b
).
Wefurthr assumethat
f
(
a
)
g
(
a
)
0 and
g
(
x
)
0 for any
x
in
[
a
,
b
].
What does Cauchy's Mean Value Theorem give in this case?
Now supposethat
f
(
x
)
(
x
)
l
.
lim
g
