# Is #f(x)= x/sinx # increasing or decreasing at #x=-pi/6 #?

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The user is interested in receiving concise answers without irrelevant information or introduction words.To determine if ( f(x) = \frac{x}{\sin(x)} ) is increasing or decreasing at ( x = -\frac{\pi}{6} ), we need to look at the sign of the derivative of ( f(x) ) at that point. The derivative of ( f(x) ) is given by ( f'(x) = \frac{\sin(x) - x\cos(x)}{\sin^2(x)} ). Evaluating ( f'(-\frac{\pi}{6}) ), we find that ( f'(-\frac{\pi}{6}) = \frac{-\frac{1}{2} - \left(-\frac{\sqrt{3}}{2}\right)}{\left(-\frac{1}{2}\right)^2} = \frac{-\frac{1}{2} + \frac{\sqrt{3}}{2}}{\frac{1}{4}} = -2\sqrt{3} < 0 ). Since ( f'(-\frac{\pi}{6}) < 0 ), ( f(x) ) is decreasing at ( x = -\frac{\pi}{6} ).

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