The Real And Complex Number Systems

f x fx
x
So, let x y, wehave
f x
fx f0
x 0
f x f 0where 0, x. *
gx gy g zx y, wherey z x
f zz fz
x y
0by(*)
which implies that g is an increasing function on 0, 1.
5.18 Assume f has a finite derivative in a, b and is continuous on a, b with
fa fb 0. Prove that for every real there is some c in a, b such that
f c fc.
Hint. Apply Rolle’s Theorem to gxfx for a suitable g depending on .
Proof: Consider gx fxe x , then by Rolle’s Theorem,
ga gb g ca b, wherec a, b
0
which implies that
f c fc.
z 2
Remark: (1) The finding of an auxiliary function usually comes from the equation that
we consider. We will give some questions around this to get more.
(2)There are some questions about finding auxiliary functions; we write it as follows.
(i) Show that e e .
Proof: (STUDY) Since log x is a strictly increasing on 0, , in order to show
e e , it suffices to show that
log e log e log e e log
which implies that
log e
Consider fx logx
x
e
: e, , wehave
f x 1 log x
x 2
log
.
0wherex e, .
So, we know that fx is strictly decreasing on e, . Hence,
e e .
(ii) Show that e x 1 x for all x R.
loge
e
log
.Thatis,
Proof: By Taylor Theorem with Remainder Term, we know that
e x 1 x ec
2 x2 ,forsomec.
So, we finally have e x 1 x for all x R.
Note: (a) The method in (ii) tells us one thing, we can give a theorem as follows. Let
f C 2n1 a, b, andf 2n x exists and f 2n x 0ona, b. Then we have
2n1
fx f
k
a
.
k!
k0