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The funciton $f (x) = \cos (\frac{π}{x})$ is monotonically increasing in the interval (K is any positive integer) is

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$(\frac{1}{2 K + 1}, \frac{1}{2 K + 2})$

$(\frac{1}{2 K + 1}, \frac{1}{2 K})$

$(\frac{1}{2 K}, \frac{1}{2 K + 1})$

$(\frac{1}{2 K + 2}, \frac{1}{2 K + 1})$

answer is A.

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Detailed Solution

$f (x) = \cos (π / x) f^{I} (x) = - \sin (π / x) (- π / x^{2})$

$f (x)$ is increasing

$\Rightarrow f^{I} (x) > 0 \sin (π / x) (π / x^{2}) > 0 \sin (π / x) > 0 π / x \in (2 kπ, 2 kπ + π) 1 / x \in (2 k, 2 k + 1) x \in (1 / 2 k + 1, 1 / 2 k)$

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