Question

Let $f :R→R$ be a continuous function such that $f(3x) – f(x) = x$. If $f(8) = 7$, then $f(14$) is equal to

• A. 4
• B. 10
• C. 11
• D. 18

Answer 1

Answer: (B)

10

Answer 2

$f(3x)–f(x)=x$ …(1)

$x→\frac x3$

$f(x)−f(\frac x3)=\frac x3$ ⋯(2)

Again $x→\frac x3$

$f(\frac x3)−f(\frac x9)=\frac {x}{3^2}$ ⋯(3)

Similarly $f(\frac {x}{3^{n−2}})−f(\frac {x}{3^{n−1}})=\frac {x}{3^{n−1}}$ ⋅⋅⋅⋅⋅⋅⋅(n)

Adding all these and applying $n→∞$

$\lim\limits _{n→∞}(f(3x)−f(\frac {x}{3^{n−1}}))=x(1+\frac 13+\frac {1}{3^2}+⋯)$

$f(3x)−f(0)=\frac {3x}{2}$

Putting $x=\frac 83$
$f(8) – f(0) = 4$
$⇒ f(0) = 3$
Putting $x=\frac {14}{3}$
$f(14)–3=7$
$f(14)=10$

So, the correct option is (B): $10$