If `f:RrarrR` is a continuous function satisfying `f(0)=1` and `f(2x)-f(x)=xAAxepsilonR` and `lim_(nrarroo)(f(x)-f(x/(2^(n))))=P(x)`. Then `P(x)` is

Options

A. a constant function
B. a linear polynomial in x
C. a quadratic polynomial in x
D. a cubic polynomial in x

Answer

B

Explanation

We have `f(2x)-f(x)=x`
`Rightarrow f(x)-f((x)/(2))=(x)/(2)" "["Replacing x by"(x)/(2)]`
`Rightarrow f((x)/(2))-f((x)/(4))=(x)/(4)`
`Rightarrow f((x)/(4))-f((x)/(8))=(x)/(8)`
`f((x)/(2^(n-1)))-f((x)/(2^(n)))=(x)/(2^(n))`
Adding up all of these questtions, we obtain
`f(x)-f((x)/(2^(n)))=(x)/(2)+(x)/(4)+(x)/(8)+.....+(x)/(2^(n))`
`Rightarrow f(x)-f((x)/(2^(n)))=(x)/(2) ((1-(1)/(2^(n))))/((1-(1)/(2)))`
`Rightarrow f(x)-f((x)/(2^(n)))=x (1-(1)/(2^(n)))`
`Rightarrow underset(n to oo)lim {f(x)-f((x)/(2^(n)))}=x`
`P(x)=x,` which is linear polynomial in x.