The Real And Complex Number Systems

h h
x x 1
lim lim
h 2 x 2 o h x 2
h0 h0 h h
x x 1 h 2 x 2 o h x 2
lim
h0
1
lim
2 h x 2 o h x 2
h x 2 1 2 h x 3 o h x 3
o1
2
1 1 h 2 x o h x
h0
1
1/2.
5.12 Take fx 3x 4 2x 3 x 2 1andgx 4x 3 3x 2 2x in Theorem 5.20. Show
that f x/g x is never equal to the quotient f1 f0/g1 g0 if 0 x 1. How
do you reconcile this with the equation
fb fa
gb ga
obtainable from Theorem 5.20 when n 1
Solution: Note that
12x 2 6x 2 12 x 1
4
11
48
f x 1
g x 1 , a x 1 b,
x 1
4
11
48
, where (0 1 4 11
48 1).
So, when we consider
f1 f0
g1 g0 0
and
f x 12x 3 6x 2 2x xg x x12x 2 6x 2,
we CANNOT write f x/g x x. Otherwise, it leads us to get a contradiction.
Remark: It should be careful when we use Generalized Mean Value Theorem, we
had better not write the above form unless we know that the denominator is not zero.
5.13 In each of the following special cases of Theorem 5.20, take n 1, c a, x b,
and show that x 1 a b/2.
(a) fx sin x, gx cosx;
Proof: Since, by Theorem 5.20,
sin a sin b sinx 1 2cos a b sin
2
a 2
b sinx 1
cosa cosbcosx 1
2sin a 2
b sin a 2
b cosx 1 ,
we find that if we choose x 1 a b/2, then both are equal.
(b) fx e x , gx e x .
Proof: Since, by Theorem 5.20,
e a e b e x 1 e
a
e b e x 1 ,
we find that if we choose x 1 a b/2, then both are equal.
Can you find a general class of such pairs of functions f and g for which x 1 will always
be a b/2 and such that both examples (a) and (b) are in this class