Let f(x) = x sin \pix, x > 0. Then for all natural numbers n, f '(x) vanishes at

Engineering

Mathematics

Monotonicity

Question

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

A unique point in the interval $(n, n + \frac{1}{2})$

a unique point in the interval $(n + \frac{1}{2}, n + 1)$

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

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

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f (x) = x sin $π$ x ; x > 0

f ' (x) = sin $π$ x + $π$ x cos $π$ x

f ' (x) = 0 $\Rightarrow$ tan ( $π$ x) = – ( $π$ x)

similar like, tan $θ$ = – $θ$

Graph are intersecting in $(\frac{1}{2}, 1) and (\frac{3}{2}, 2)$

option (B) and (C) are satisfying.