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JEE Advanced2013MathematicsDifferentiationActual

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

Options

  1. AA unique point in the interval n , n + 1 2
  2. BA unique point in the interval n + 1 2 , n + 1
  3. CA unique point in the interval (n, n + 1)
  4. DTwo points in the interval (n, n + 1)

Correct answer

B. A unique point in the interval n + 1 2 , n + 1

Step-by-step 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 tanπ x & - πx . And their intersection points will be the points, where f ' x vanishes.

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