Question

Which one of the following option is always correct for $x>0$

• A. $e^x\le 1+x$
• B. $e^x<1+x$
• C. $e^x\ge 1+x$
• D. $e^x>1+x$

Answer 1

The correct option is D $e^x>1+x$

Let $f\left(x\right)=e^x-1-x$
Now for critical points, $f^{\prime }\left(x\right)=e^x-1=0\Rightarrow x=0$
For $x>0$; $f^{\prime }\left(x\right)>0$ i.e. strictly increasing and
For $x<0$; $f^{\prime }\left(x\right)<0$ i.e. strictly decreasing
So, $f\left(x\right)>f\left(0\right)$ for $x>0$
$\Rightarrow e^x-1-x>1-1-0$
$\Rightarrow e^x>1+x$