i)     By calculation, we have
P = æ
ç
è 		ta+b+1
a+b+1
 ö
÷
ø 		2

 
 = t2(a+b+1)
(a+b+1)2
 ,
and
Q = æ
ç
è 		t2a+1
2a+1
 ö
÷
ø 		æ
ç
è 		t2b+1
2b+1
 ö
÷
ø 		= t2(a+b+1)
(2a+1)(2b+1)
 .
Thus we must decide which of the two expressions
(a+b+1)2,        (2a+1)(2b+1)
is the smallest.

Since
(a+b+1)2-(2a+1)(2b+1) = (a-b)2 ³ 0
we see that P £ Q.
(ii)     By computing [f(x)+lg(x)]2 and integrating, we have
U + 2Vl+ Wl2 ³ 0,
(*)
for all l, where
U = ó
õ 		t

0 
 [f(x)]2 dx,    V = ó
õ 		t

0 
 f(x)g(x) dx,    W = ó
õ 		t

0 
 [g(x)]2 dx.
The condition that (*) holds for all l is UW ³ V2 which is the required inequality.