Application of Derivatives 2 Question 6
6. Let $f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}, x \in R$, where $a, b$ and $d$ are non-zero real constants. Then,
(2019 Main, 11 Jan II)
(a) $f$ is an increasing function of $x$
(b) $f^{\prime}$ is not a continuous function of $x$
(c) $f$ is a decreasing function of $x$
(d) $f$ is neither increasing nor decreasing function of $x$
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Solution:
- We have,
$f(x)=\frac{x}{\left(a^{2}+x^{2}\right)^{1 / 2}}-\frac{(d-x)}{\left(b^{2}+(d-x)^{2}\right)^{1 / 2}}$
Differentiating above w.r.t. $x$, we get
$f^{\prime}(x)=\frac{\left(a^{2}+x^{2}\right)^{1 / 2}-x \frac{1}{2} \frac{2 x}{\left(a^{2}+x^{2}\right)^{1 / 2}}}{\left(a^{2}+x^{2}\right)}$
$$ -\frac{\left(b^{2}+(d-x)^{2}\right)^{1 / 2}(-1)-(d-x) \frac{2(d-x)(-1)}{2\left(b^{2}+(d-x)^{2}\right)^{1 / 2}}}{\left(b^{2}+(d-x)^{2}\right)} $$
[by using quotient rule of derivative]
$$ \begin{aligned} = & \frac{a^{2}+x^{2}-x^{2}}{\left(a^{2}+x^{2}\right)^{3 / 2}}+\frac{b^{2}+(d-x)^{2}-(d-x)^{2}}{\left(b^{2}+(d-x)^{2}\right)^{3 / 2}} \\ = & \frac{a^{2}}{\left(a^{2}+x^{2}\right)^{3 / 2}}+\frac{b^{2}}{\left(b^{2}+(d-x)^{2}\right)^{3 / 2}}>0, \\ & \forall x \in R \end{aligned} $$
Hence, $f(x)$ is an increasing function of $x$.
