First slide

Functions (XII)

Question

If $f . (0, π) \to R$ is given by $f (x) = \sum_{k = 1}^{n} [1 + \sin k x], [x]$ denotes the greatest integer function, then the range of $f (x)$ is

Moderate

A

${n - 1, n + 1}$
B

${n}$
C

${n, n + 1}$
D

${n - 1, n}$

Solution

$f (x) = \sum_{k = 1}^{n} 1 + [\sin k x] = n + [\sin x] + [\sin 2 x] + \dots$
$+ [\sin n x] .$ If $k x \neq \frac{π}{2}$ for any $k = 1, 2 . . . . . . . . . n$ then $0$
$[\sin k x] = 0, k = 1, 2 \dots n .$ i.e. $f (x) = n .$ If $k x = \frac{π}{2}$
for some $k$ then $x = \frac{π}{2 k}$ , hence $\sin x, \sin 2 x, \dots \sin (k - 1) x$ will lie between $0$ and $1$ so $[\sin j x] = 0$
$1 \leq j \leq k - 1; \sin k x = 1$ so $f (x)$ can be $n + 1$ or $n .$

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