Check whether the following function is strictly decreasing on $(0, \frac{π}{2})$ or not.

$c o s 3 x$

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Solution

Let f(x) =cos3x , f'(x)=-3sin3x. In interval $(0, \frac{π}{2}), f^{'} (x) < 0$
Because sin 3x will either lie in the first or second quadrant which will give a positive value. Therefore, f(x) is strictly decreasing on $(0, \frac{π}{2})$ .
When $x ϵ (\frac{π}{3}, \frac{π}{2})$ , then $f^{'} (x) > 0$
Because $s i n 3 x$ will lie in the third quadrant.
Therefore, f(x) is not strictly decreasing on $(0, \frac{π}{2})$

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