Question

For all $x\,\in\,(0,1)$

• A. $e^x,
• B. $\log_e (1 + x) < x$
• C. $\sin x > x$
• D. $\log_e x > x$

Answer 1

Answer: (B)

$\log_e (1 + x) < x$

Answer 2

Let $f\left(x\right) = e^{x} - 1 - x$ then $f'\left(x\right) =e^{x } - 1 >0$ for $x \in\left(0,1\right)$
$\therefore \, f(x)$ is an increasing function.
$\therefore \, f(x) > f(0) , \forall x \in (0,1)$
$\Rightarrow \, e^x - 1 - x > 0 \Rightarrow \, e^x > 1 + x$
$\therefore$ (a) does not hold.
(b) Let $g(x) = log (1+x) -x$
then $g'(x) = \frac{1}{1+x} - 1 = - \frac{x}{1+x} < 0 , \forall x \in (0,1)$
$\therefore \, g(x)$ is decreasing on $(0, 1) \therefore x > 0$
$\Rightarrow \, g (x) < g (0)$
$\Rightarrow \, \log_e (1 + x) - x < 0 \, \Rightarrow \, \log_e (1 + x) < x$