The Real And Complex Number Systems

fx f0 f 0x f 0
x
2!
2 f3 c
x
3!
3 ,wherec x,0 or 0, x,
Then let x 1, and subtract one from another, we get
f 3 c 1 f 3 c 2 6, where c 1 and c 2 in 1, 1.
So,wehaveprovef 3 x 3forsomex 1, 1.
5.24 If h 0andiff x exists (and is finite) for every x in a h, a h, andiff is
continuous on a h, a h, show that we have:
(a)
fa h fa h
f
h
a h f a h,0 1;
Proof: Let gh fa h fa h, then by Mean Vaule Theorem, wehave
gh g0 gh
g hh, where 0 1
f a h f a hh
which implies that
fa h fa h
f
h
a h f a h,0 1.
(b)
fa h 2fa fa h
f
h
a h f a h,0 1.
Proof: Let gh fa h 2fa fa h, then by Mean Vaule Theorem, wehave
gh g0 gh
g hh, where 0 1
f a h f a hh
which implies that
fa h 2fa fa h
f
h
a h f a h,0 1.
(c) If f a exists, show that.
f fa h 2fa fa h
a lim
h0 h 2
Proof: Since
fa h 2fa fa h
lim
h0 h 2
f
lim
a h f a h
by L-Hospital Rule
h0 2h
lim
h0
f a h f a
2h
1 2 2f a since f a exists.
f a.
f a f a h
2h
Remark: There is another proof by using Generalized Mean Value theorem.
Proof: Let g 1 h fa h 2fa fa h and g 2 h h 2 , then by Generalized
Mean Value theorem, we have