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)`

Options

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)`

Answer

A and C

Explanation

`"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)`