Question

Which of the following function is a monotonic increasing function for all values of $x$

• A. $\sin x$
• B. $\cos x$
• C. $e^x$
• D. $2^{-x}$

Answer 1

The correct option is C $e^x$

$f\left(x\right)=\sin x$
$f^{\prime }\left(x\right)=\cos x$
Now
$f^{\prime }\left(x\right)<0$
Hence
$\cos \left(x\right)<0$
$xϵ(\frac{\pi }{2},\frac{3\pi }{2})$.
Similarly for $f\left(x\right)=\cos x$.
Now
$f\left(x\right)=e^x$
$f^{\prime }\left(x\right)=e^x$
And
$e^x>0$ for all real $x$.
Hence
$f^{\prime }\left(x\right)>0$ for all real $x.$
Hence
$f\left(x\right)$ is an increasing function for all real values of $x.$
Consider
$f\left(x\right)=2^{-x}$
$f^{\prime }\left(x\right)$
$=-2^{-x}$
Now
$f^{\prime }\left(x\right)<0$ for all real values of $x.$
Hence it is a decreasing function for all real values of $x.$