Let `f(x)=xsinpix ,x > 0.` Then for all natural numbers `n ,f^(prime)(x)` vanishes at a unique point in the interval `(n , n+1/2)` a unique point in the interval `(n+1/2, n+1)` a unique point in the interval `(n , n+1)` two points in the interval `(n , n+1)`

Explanation

We have `f'(x)=sinpix+pixcospix=0`
`implies" "tanpix=-pix`
Now we draw the graphs of `y=tanpix` and `y=-pix` and look for their points of intersection. `y=tanpix` has period 1.
Graphs are as shown in the following figure.

`implies" " x in (n + (1)/(2), n + 1) or (n, n + 1)`
Hence (b) and (c) are correct options.