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Let f(x)=x sin pix, x gt 0 Then for all ...

Let `f(x)=x sin pix`, `x gt 0` Then for all natural numbers n, f`(x) vanishes at

A

`"a unique point in the interval "(n,n+(1)/(2))`

B

`"a unique point in the interval "(n+(1)/(2),n+1)`

C

`"a unique point in the interval "(n,n+1)`

D

`"two points in the interval "(n,n+1)`

Text Solution

Verified by Experts

`"We have "f'(x)=sin pix+pi x cos pi x=0`
`"or "tan pi x=-pix`
The graph of `y=tan pi x and y= - pi x` is as shown in the following figure. Therefore,
`tan pix=-pix`

From the graph
`x in (n+(1)/(2),n+1)or (n, n+1)`
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