QJEEM22Maac4e: JEE Main 2022 Mathematics - Class 12 JEEM22Maac4e Solution: Let f : R R be a continuous function such that . If , then is…

$f(3x) - f(x) = x$ ...... (1)

$x \to {x \over 3}$

$f(x) - f\left( {{x \over 3}} \right) = {x \over 3}$ ....... (2)

Again $x \to {x \over 3}$

$f\left( {{x \over 3}} \right) - f\left( {{x \over 9}} \right) = {x \over {{3^2}}}$ ...... (3)

Similarly

$f\left( {{x \over {{3^{n - 2}}}}} \right) - f\left( {{x \over {{3^{n - 1}}}}} \right) = {x \over {{3^{n - 1}}}}\,.....\,(n)$

Adding all these and applying $n \to \infty$

$\mathop {\lim }\limits_{n \to \infty } \left( {f(3x) - f\left( {{x \over {{3^{n - 1}}}}} \right)} \right) = x\left( {1 + {1 \over 3} + {1 \over {{3^2}}}\, + \,....} \right)$

$f(3x) - f(0) = {{3x} \over 2}$

Putting $x = {8 \over 3}$

$f(8) - f(0) = 4$

$\Rightarrow f(0) = 3$

Putting $x = {{14} \over 3}$

$f(14) - 3 = 7 \Rightarrow f(14) = 10$