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

Answer 1

#f(x)# is 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.

To answer, then, we first must find the derivative of #f(x)#, then evaluate it at #x=pi/6#, and finally check the sign of the result.
First, let's use the quotient rule to evaluate the derivative: #f'(x) = d/dxcos(x)/e^x#
# = (-sin(x)e^x - cos(x)e^x)/(e^x)^2#
#=(-e^x(cos(x)+sin(x)))/(e^x)^2#
#= -(cos(x)+sin(x))/e^x#
Next, we evaluate the first derivative at #x=pi/6#
#f'(pi/6) = -(cos(pi/6)+sin(pi/6))/e^(pi/6)#
#= -(sqrt(3)/2 + 1/2)/e^(pi/6)#
#= (-sqrt(3)-1)/(2e^(pi/6))#

Lastly, we verify the outcome's sign.

#-sqrt(3) - 1 < 0# and #2e^(pi/6) > 0#

Consequently, since a positive number and a negative number have a negative quotient,

#f'(pi/6) < 0#
meaning #f(x)# is decreasing at #x=pi/6#.
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Answer 2

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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Answer from HIX Tutor

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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