Question

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

• A. $e^x<1+x$
• B. ${\log }_e(1+x)<x$
• C. $\sin x>x$
• D. ${\log }_{e}x>x$

Answer 1

The correct option is B ${\log }_e(1+x)<x$

$y=e^x\Rightarrow \frac{dy}{dx}=e^x$
Thenm equation of the tangent at $x=0$ is $y-1=1(x-0)$ or $y=x+1$
Graph of $y=e^x$ always lies above the graph of $y=1+x$,
Hence, $e^x>1+x\Rightarrow x>{\log }_e(1+x)$. Hence, $b$ is true.