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:

(A) f is an increasing function of x

(B) f is a decreasing function of x

(C) f is not a continuous function of x

(D) f is neither increasing nor decreasing function of x

JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Application of Derivatives

Let

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

x \in R

, where

a, b

and

d

are non-zero real constants. Then:

Answer: (A)

Differentiating,

f^{'} (x) = \frac{a ^{2}}{( x ^{2} + a ^{2} ) ^{3/2}} + \frac{b ^{2}}{( b ^{2} + ( d - x ) ^{2} ) ^{3/2}}

. Both terms are positive for every real

x

, since each numerator and denominator is positive, so

f^{'} (x) > 0

throughout and

f

is an increasing function of

x

JEE Main · 2019 · 11 Jan 2019 · Shift 2 · Single correct option