If the solve the problem

Question:

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

where a, b and d are non-zero real constants. Then :-

  1. $\mathrm{f}$ is a decreasing function of $x$

  2. $\mathrm{f}$ is neither increasing nor decreasing function of $x$

  3. $f^{\prime}$ is not a continuous function of $x$

  4. $\mathrm{f}$ is an increasing function of $x$

Correct Option: , 4

Solution:

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

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

$f(x)$ is an increasing function.

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