Question

Let $f\left(x\right)=\ln (1+x)-x,x>0.$ Then which of the following is correct

• A. $f\left(x\right)>0$
• B. $f\left(x\right)\ge 0$
• C. $f\left(x\right)<0$
• D. $f\left(x\right)\le 0$

Answer 1

The correct option is C $f\left(x\right)<0$

Given : $f\left(x\right)=\ln (1+x)-x,x>0.$
Differentiating the above w.r.t. $x$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{1+x}-1=\frac{-x}{1+x}$
$\Rightarrow f^{\prime }\left(x\right)<0\forall x>0$
$\therefore f\left(x\right)$ is monotonically decreasing.
and $f\left(0\right)=0$
Since $x>0\Rightarrow f\left(x\right)<f\left(0\right)$
$\therefore f\left(x\right)<0$