Question

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

A. p

B. p-1

C. 2p+1

D. 2p-1

Answer 1

[n + p sin x] = n + [p sin x].

The points of discontinuity and hence of non-differentiability are the points where p sin x is an integer. There are no other points of non-differentiability.

So, the points of discontinuity are when sin x = 1/p or 2/p,..., (p-1)/p each of which have two corresponding values of x and sin x = 1, which has a unique solution in the given interval.
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That makes 2(p-1)+1 = 2p-1 solutions