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If f(x)=[2x]sin3pix then prove that f'(k...

If `f(x)=[2x]sin3pix` then prove that `f'(k^(+))=6kpi(-1)^(k)`, (where [.] denotes the greatest integer function and `k in N).`

Text Solution

Verified by Experts

`f'(k)=underset(hrarr0)lim(f(k+h)-f(k))/(h)`
`=underset(hrarr0)lim([2k+2h]sin(3pi(k+h))-[2k]sin(3kpi))/(h)`
`=underset(hrarr0)lim([2k]sin(3kpi+3pih))/(h)" "(as sin(3kpi)=0)`
`=2kxxunderset(hrarr0)lim((-1)^(k)sin(3pih))/(h)" (as 2k is integer)"`
`=2kxx(-1)^(k)xx(3pi)underset(hrarr0)lim(sin(3pih))/((3pih))`
`=6kpi(-1)^(k)`
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