Let f ( x ) = x sin ( x ) , x 0 . Then for all natural numbers n , f ( x ) vanishes at [2013]

Question

Let $f (x) = x \sin (π x), x > 0$ . Then for all natural numbers $n, f' (x)$ vanishes at [2013]

Smart Achievers · Accepted Answer

(2, 3)

Given: $f (x) = x \sin π x, x > 0$

$\Rightarrow f' (x) = \sin π x + x π \cos π x$

$Now, f' (x) = 0 \Rightarrow \tan π x = - π x$

$From graph of y = \tan π x and y = - π x, it is clear that they intersect each other at a unique point in the intervals$

$(n, n + 1) and (n + \frac{1}{2}, n + 1)$

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