1.

Let f(x)={(xsin((pi)/x), "at" xgt0),(0,"at"x=0):}

Answer»

`f^(')(x)` VANISHES atleast once in `[1/3,1/2]`
`f^(')(x)` vanishes atleast once in `[1/4,1/2]`
`f^(')(x)` satisfies Roll's THEOREM on `[0,1]`
`f^(')(x)` vanishes atleast once in `[1/(k+1),1/k]` for every `KepsilonN`

Solution :`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`


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