Question

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

• A. $e^x<1+x$
• B. $lo{g}_e(1+x)<x$
• C. $sinx>x$
• D. $lo{g}_{e}x>x$

Answer 1

The correct option is B $lo{g}_e(1+x)<x$

Both $e^x$ and 1+x are increasing and $\sqrt{e}\ge 1+\frac{1}{2}$, because $\sqrt{e}=1.65$ nearly. so the answer (a) is not correct. Since $sin\frac{\pi }{6}<\frac{\pi }{6}$ becuase $\frac{1}{2}<\frac{22}{42}$.So, (c) is not correct.$log\frac{1}{2}<\frac{1}{2}$ because $log\frac{1}{2}$ is negative.
$\therefore$ Option (d) is not correct.
Thus, by elimination (b) is correct.