Let f ⁡ x = x  sin  π x , x > 0 . Then for all natural numbers n ,  f ⁡…

Let  f⁡ x = x  sin  π x ,  x > 0 . Then for all natural numbers n ,  f⁡ ′ x  vanishes at
  1. A unique point in the interval  n ,  n + 1 2
  2. A unique point in the interval  n + 1 2 ,  n + 1
  3. A unique point in the interval (n, n + 1)
  4. Two points in the interval (n, n + 1)

Solution

f x = x sin  π x , x > 0

As f'x vanishes,
f ′ x = sin  π x + π x cos π x = 0
tan π x = - π x

We will draw the graph of tanπx & -πx.

And their intersection points will be the points, where f'x vanishes.

Asked in: JEE Advanced 2013 (Paper 1)

Practice more Differentiation questions on Aicharya