Let $f : R \rightarrow R$ be a continuous function such that $f (3x)- f(x)= x$. If $f(8)=7$, then $f (14)$ is equal to :
- 4
- 10
- 11
- 16
The Correct Option is B
Solution and Explanation
The correct option is (B) : 10
\(f(x)-f(\frac{x}{3})=\frac{x}{3}\)
\(f(\frac{x}{3})-f(\frac{x}{3^2})=\frac{x}{3^2}\)
By adding this …...
\(f(x)-\lim\limits_{n\rightarrow\infty}f(\frac{x}{3^n})=x(\frac{1}{3}+\frac{1}{3^2}.....\infty)\)
\(f(x)-f(0)=\frac{x}{2}\)
\(f(8)=7;f(0)=3\)
\(f(x)=\frac{x}{2}+3\)
\(f(14)=10\)
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.