Is #f(x)=cosx/e^x# increasing or decreasing at #x=pi/6#?
If the first derivative of a function is positive or negative when evaluated at a particular point, the function is said to be increasing or decreasing at that point.
Lastly, we verify the outcome's sign.
Consequently, since a positive number and a negative number have a negative quotient,
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To determine whether ( f(x) = \frac{\cos(x)}{e^x} ) is increasing or decreasing at ( x = \frac{\pi}{6} ), we need to evaluate the derivative of ( f(x) ) at that point. If the derivative is positive, the function is increasing; if negative, it's decreasing.
First, find the derivative of ( f(x) ): [ f'(x) = \frac{-\sin(x)e^x - \cos(x)e^x}{e^{2x}} ]
Evaluate ( f'(x) ) at ( x = \frac{\pi}{6} ): [ f'\left(\frac{\pi}{6}\right) = \frac{-\sin\left(\frac{\pi}{6}\right)e^{\frac{\pi}{6}} - \cos\left(\frac{\pi}{6}\right)e^{\frac{\pi}{6}}}{e^{\frac{\pi}{3}}} ]
[ = \frac{-\frac{1}{2} \cdot e^{\frac{\pi}{6}} - \frac{\sqrt{3}}{2} \cdot e^{\frac{\pi}{6}}}{e^{\frac{\pi}{3}}} ]
[ = \frac{-\frac{1}{2} - \frac{\sqrt{3}}{2}}{e^{\frac{\pi}{3}}} ]
[ = \frac{-\left(\frac{1 + \sqrt{3}}{2}\right)}{e^{\frac{\pi}{3}}} ]
[ = \frac{-(1 + \sqrt{3})}{2e^{\frac{\pi}{3}}} ]
Since the numerator is negative and the denominator is always positive, ( f'\left(\frac{\pi}{6}\right) ) is negative.
Therefore, ( 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.
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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