Is #f(x)=(x-e^x)/(x-1)^3# increasing or decreasing at #x=2#?

Answer 1

Increasing.

To know if a function #f(x)# is increasing or decreasing at some point, you must find the derivative #f'(x)# of the function and check the sign of the value of the derivative. If #f'(x) > 0#, the function is said to be increasing. If #f'(x) < 0#, the function is said to be decreasing.
Begin by using the Quotient Rule to find the derivative #f'(x)#:
#f'(x) = ( (x-1)^3 * (1 - e^x) - (x - e^x) * (3 * (x-1)^2) ) / ((x-1)^3)^2 # # = ( (x-1)^3 * (1 - e^x) - 3(x - e^x)(x-1)^2 ) / (x-1)^6 #
Now we can evaluate #f'(2)#:
#f'(2) = ( (2-1)^3 * (1 - e^2) - 3(2 - e^2)(2-1)^2 ) / (2-1)^6 # # = ( (1 - e^2) - 3(2 - e^2) ) / 1 = (1 - e^2 - 6 +3e^2 ) # # = 2e^2 - 5 ~~ 9.778 > 0#
Since #f'(2) > 0#, we say #f(x)# is increasing at #x = 2#.

This can also be verified by looking at the graph as a support for our work:

graph{(x-e^x)/(x-1)^3 [-5.484, 12.296, -7.46, 1.425]}

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Answer 2

To determine if the function ( f(x) = \frac{x - e^x}{(x - 1)^3} ) is increasing or decreasing at ( x = 2 ), we need to evaluate the sign of its derivative at that point.

The derivative of ( f(x) ) with respect to ( x ) can be found using the quotient rule:

[ f'(x) = \frac{(x - 1)^3 \cdot (1 - e^x) - (x - e^x) \cdot 3(x - 1)^2}{(x - 1)^6} ]

Evaluating this derivative at ( x = 2 ) gives:

[ f'(2) = \frac{(2 - 1)^3 \cdot (1 - e^2) - (2 - e^2) \cdot 3(2 - 1)^2}{(2 - 1)^6} ]

[ f'(2) = \frac{(1) \cdot (1 - e^2) - (2 - e^2) \cdot 3(1)}{(1)^6} ]

[ f'(2) = \frac{1 - e^2 - 6 + 3e^2}{1} ]

[ f'(2) = -5 + 2e^2 ]

Since ( f'(2) = -5 + 2e^2 > 0 ), the function 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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