# Is #f(x)= cot(3x+(5pi)/8) # increasing or decreasing at #x=pi/4 #?

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To determine if ( f(x) = \cot\left(3x + \frac{5\pi}{8}\right) ) is increasing or decreasing at ( x = \frac{\pi}{4} ), we need to examine the sign of its derivative at that point. The derivative of ( f(x) ) with respect to ( x ) is given by:

[ f'(x) = -3 \csc^2\left(3x + \frac{5\pi}{8}\right) ]

Evaluate ( f'\left(\frac{\pi}{4}\right) ). If ( f'\left(\frac{\pi}{4}\right) > 0 ), then ( f(x) ) is increasing at ( x = \frac{\pi}{4} ). If ( f'\left(\frac{\pi}{4}\right) < 0 ), then ( f(x) ) is decreasing at ( x = \frac{\pi}{4} ).

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