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1)

Which of the following inequality is true for x>0 ?


A) log(1+x)<x1+x<x

B) x1+x<x<log(1+x)

C) x<log(1+x)<x1+x

D) x1+x<log(1+x)<x

Answer:

Option D

Explanation:

Let f(x) = log(1+x)x1+x

       f(x)=11+x(1+x).1x.1(1+x)2

     =11+x1(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>0x2+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]