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

Increasing.

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To determine if ( f(x) = \frac{\cos(x)}{\pi - e^x} ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to evaluate the derivative of ( f(x) ) at that point and examine its sign.

The derivative of ( f(x) ) with respect to ( x ) is given by:

[ f'(x) = \frac{-\sin(x)(\pi - e^x) - \cos(x)(-e^x)}{(\pi - e^x)^2} ]

Evaluating ( f'(x) ) at ( x = \frac{\pi}{6} ), we have:

[ f'\left(\frac{\pi}{6}\right) = \frac{-\sin\left(\frac{\pi}{6}\right)\left(\pi - e^{\frac{\pi}{6}}\right) - \cos\left(\frac{\pi}{6}\right)\left(-e^{\frac{\pi}{6}}\right)}{\left(\pi - e^{\frac{\pi}{6}}\right)^2} ]

[ = \frac{-\frac{1}{2}\left(\pi - e^{\frac{\pi}{6}}\right) - \frac{\sqrt{3}}{2}\left(-e^{\frac{\pi}{6}}\right)}{\left(\pi - e^{\frac{\pi}{6}}\right)^2} ]

[ = \frac{-\frac{1}{2}\pi + \frac{1}{2}e^{\frac{\pi}{6}} + \frac{\sqrt{3}}{2}e^{\frac{\pi}{6}}}{\left(\pi - e^{\frac{\pi}{6}}\right)^2} ]

Now, evaluate the sign of ( f'\left(\frac{\pi}{6}\right) ) to determine if ( f(x) ) is increasing or decreasing at ( x = \frac{\pi}{6} ). If ( f'\left(\frac{\pi}{6}\right) > 0 ), then ( f(x) ) is increasing at ( x = \frac{\pi}{6} ). If ( f'\left(\frac{\pi}{6}\right) < 0 ), then ( f(x) ) is decreasing at ( x = \frac{\pi}{6} ).

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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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