Question

Let $f\left(x\right)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x{)}^2}},x\in R$,

where $a,b$ and $d$ are non-zero real constants, then :

• A. $f$ is an increasing function of $x$
• B. $f$ is neither increasing nor decreasing function of $x$
• C. $f$ is a decreasing function of $x$
• D. $f^{\prime }$ is not a continuous function of $x$

Answer 1

The correct option is A $f$ is an increasing function of $x$

$f\left(x\right)=\frac{x}{\sqrt{a^2+x^2}}-\frac{(d-x)}{\sqrt{b^2+(d-x{)}^2}}$

$f^{\prime }\left(x\right)=\frac{\sqrt{a^2+x^2}-x\cdot \frac{1}{2\sqrt{a^2+x^2}}\cdot 2x}{a^2+x^2}+\frac{\sqrt{b^2+(d-x{)}^2}-\frac{(d-x)\cdot 2\cdot (d-x)}{2\sqrt{b^2+(d-x{)}^2}}}{b^2+(d-x{)}^2}$

$=\frac{\sqrt{a^2+x^2}-\frac{x^2}{\sqrt{a^2+x^2}}}{a^2+x^2}+\frac{\sqrt{b^2+(d-x{)}^2}-\frac{(d-x{)}^2}{\sqrt{b^2+(d-x{)}^2}}}{b^2+(d-x{)}^2}$


$=\frac{a^2}{(a^2+x^2{)}^{3/2}}+\frac{b^2}{(b^2+(d-x{)}^2{)}^{3/2}}>0$

Hence, $f$ is an increasing function.