Question

if x> 0 then show that log(1+x)>(x)/(1+x)​

Answer 1

Answer:

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

for all x>0 using the mean value theorem. I tried to prove the two inequalities separately.

x1+x<log(1+x)⇔x1+x−log(1+x)<0

Let

f(x)=x1+x−log(1+x).

 

Since

f(0)=0

and

f′(x)=1(1+x)2−11+x<0