Answer:
Option D
Explanation:
Let f(x) = log(1+x)−x1+x
∴ f′(x)=11+x−(1+x).1−x.1(1+x)2
=11+x−1(1+x)2=x(1+x)2
Which is positive . [∵ x>0]
∴ f(x) is monoatomic increasing , when x>0.
⇒ f(x) > f (0)
now, f(0) = log 1-0=0
∴ f(x) >0
⇒ log(1+x)−x1+x>0
⇒ x1+x<log(1+x).....(i)
Also, for x >0,
x2>0⇒x2+x>x
⇒ x(x+1) >x
⇒ x>xx+1 ...........(ii)
From eqs. (i) and (ii) , we get
xx+1<log(1+x)<x
[ ∵ log(1+x)<x for x >0]