Is #f(x)=(x-2)/e^x # increasing or decreasing at #x=-2 #?

Answer 1

Increasing. See explanation below.

In order to find if a function is increasing or decreasing, we take its derivative and evaluate it at the #x#-value in question. If the derivative is positive at that point, the function is increasing; if the derivative is negative, the function is decreasing.
Step 1: Find the Derivative Since we have #x-2# divided by #e^x#, we need to use the quotient rule to take the derivative, which states: #d/dx(u/v) = (u'v-uv')/v^2# Where #u# and #v# are functions of #x#.
In our case, we have #u=x-2# and #v=e^x#. Taking the derivative of these two: #u' = 1# #v' = e^x#
Now we can substitute into the quotient rule: #d/dx((x-2)/e^x) = ((x-2)'(e^x)-(x-2)(e^x)')/(e^x)^2# #= (1(e^x)-(x-2)(e^x))/(e^(2x)#
We can do a little simplifying here: #= (e^x(1-(x-2)))/(e^(2x)# #= (1-x+2)/(e^x)# #= (-x+3)/(e^x)#
Step 2: Evaluate We are being asked to find if the function is increasing or decreasing at #x=-2#; that means we evaluate the derivative at #x=-2#: #=(-(-2)+3)/(e^(-2))# #=(5)/(e^(-2))=5e^2#
We don't even need to find #5e^2#, because it is definitely positive. And because it's positive, we can say that the function #(x-2)/e^x# is increasing at #x=-2#. To confirm, take a look at the graph of #(x-2)/e^x# and you will see it increasing at #x=-2#. graph{(x-2)/e^x [-10, 10, -5, 5]}
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Answer 2

To determine if ( f(x) = \frac{x - 2}{e^x} ) is increasing or decreasing at ( x = -2 ), we need to evaluate the derivative of ( f(x) ) at that point.

First, find the derivative of ( f(x) ): [ f'(x) = \frac{(1)(e^x) - (x - 2)(e^x)}{(e^x)^2} ] [ f'(x) = \frac{e^x - xe^x + 2e^x}{e^{2x}} ] [ f'(x) = \frac{(3 - x)e^x}{e^{2x}} ]

Now, evaluate ( f'(-2) ): [ f'(-2) = \frac{(3 - (-2))e^{-2}}{e^{-4}} ] [ f'(-2) = \frac{5e^{-2}}{e^{-4}} ] [ f'(-2) = 5e^2 ]

Since ( f'(-2) ) is positive, ( f(x) ) is increasing at ( x = -2 ).

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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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