Consider the function `f(x)={{:(xsin(pi)/(x),"for"xgt0),(0,"for"x=0):}`
Then, the number of points in `(0,1)` where the derivative f'(x) vanishes is

Options

A. `f^(')(x)` vanishes atleast once in `[1/3,1/2]`
B. `f^(')(x)` vanishes atleast once in `[1/4,1/2]`
C. `f^(')(x)` satisfies Roll's theorem on `[0,1]`
D. `f^(')(x)` vanishes atleast once in `[1/(k+1),1/k]` for every `KepsilonN`

Answer

A, B, D

Explanation

`f(x)` satisfies Rool's theorem on `[1/(k+1),1/k],KepsilonN`
`:.f^(')(x)=0` on each interval `[1/(k+1),1/k],KepsilonN`