# Is #f(x)= cot(-x+(5pi)/6) # increasing or decreasing at #x=pi/4 #?

Increasing

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To determine whether ( f(x) = \cot\left(-x + \frac{5\pi}{6}\right) ) is increasing or decreasing at ( x = \frac{\pi}{4} ), we need to analyze the sign of its derivative ( f'(x) ) at that point.

First, find the derivative of ( f(x) ) using the chain rule:

[ f'(x) = \frac{d}{dx} \cot\left(-x + \frac{5\pi}{6}\right) ] [ = -\csc^2\left(-x + \frac{5\pi}{6}\right) \cdot \frac{d}{dx} \left(-x + \frac{5\pi}{6}\right) ] [ = -\csc^2\left(-x + \frac{5\pi}{6}\right) \cdot (-1) ]

Now, evaluate ( f'(x) ) at ( x = \frac{\pi}{4} ):

[ f'\left(\frac{\pi}{4}\right) = -\csc^2\left(-\frac{\pi}{4} + \frac{5\pi}{6}\right) \cdot (-1) ]

Since ( \csc^2\left(-\frac{\pi}{4} + \frac{5\pi}{6}\right) ) is positive (since ( \csc ) is positive in the second and third quadrants), ( f'\left(\frac{\pi}{4}\right) ) is negative.

Therefore, ( f(x) ) is decreasing at ( x = \frac{\pi}{4} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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