Let f(x) = [ n + p sin x], `x in (0,pi), n in Z`, p is a prime number and [x] = the greatest integer less than or equal to x. The number of points at which f(x) is not not differentiable is :

Options

A. p
B. p-1
C. 2p+1
D. 2p-1

Answer

D

Explanation

We know that the greatest integer function [x] is neither continuous nor differentiable at integer points. Therefore, f(x)=[n+p sin x] is discontinuous and non-differentiable at those points where `n+p sinx` is an integer.
Clearly, n+p sin x will be an integer when p sin x is an integer Now, p sin x is an integer, if
`sin x=1,(1)/(p),(2)/(p),(3)/(p),........,(p-1)/(p) [therefore x in (0, pi) therefore sin x gt 0]`
`Rightarrow x=sin^(-1)(1), sin^(-1)((2)/(p)),...... sin^(-1)((p-1)/(p))` at points
`pi-"sin"^(-1)(1)/(p),....mpi-sin^(-1)((p-1)/(p))"also"`
Hence, the total number of points of discontiniuity of f(x) is (2p-1).