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)

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