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

The Correct Option is B

\(f(3x)–f(x)=x\)     …(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): \(10\)