Question

Find the intervals of monotonicity of the function : $y=\ln (x+\sqrt{1+x^2})$

• A. Increases monotonically
• B. Decreases monotically
• C. Does not change
• D. None

Answer 1

Answer: (A)

Increases monotonically

Given, $y=\ln \left(\sqrt{1+x^2}\right)$

$y^{\prime }$ $=\frac{1}{x+\sqrt{1+x^2}}.(1+\frac{x}{\sqrt{1+x^2}})$
$=\frac{1}{\sqrt{1+x^2}}$
Now for all real values of $x$, $\sqrt{1+x^2}>0$
$\frac{1}{\sqrt{1+x^2}}>0$
$f^{\prime }\left(x\right)>0$
Hence, $f\left(x\right)$ is monotonically increasing for all $xϵR$.