Question

Let $f\left(x\right) = \frac{x}{\sqrt{a^{2} + x^{2}} } - \frac{d-x}{\sqrt{b^{2} + \left(d-x\right)^{2}}} , x \in R$ where $a, b$ and $d$ are non-zero real constants. Then :

• A. $f$ is a decreasing function of $x$
• B. $f$ is neither increasing nor decreasing function of $x$
• C. $f$' is not a continuous function of $x$
• D. $f$ is an increasing function of $x$

Answer 1

Answer: (D)

$f\left(x\right) =\frac{x}{\sqrt{a^{2}+x^{2} }} - \frac{d-x}{\sqrt{b^{2}+\left(d-x\right)^{2}}} $
$ f'\left(x\right) =\frac{a^{2}}{\left(a^{2}+x^{2}\right)^{3/2}} + \frac{b^{2}}{\left(b^{2}+\left(d-x\right)^{2}\right)^{3/2}} > 0 \forall x\in R $