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

Increasing.

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]}

By signing up, you agree to our Terms of Service and Privacy Policy

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 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you determine all values of c that satisfy the mean value theorem on the interval [0,3] for #f(x) = x^3 + x - 1 #?
- What are the critical values, if any, of #f(x)=(x-5)/(x^2+8)#?
- Is #f(x)=(2x^3-7x^2+3x+1)/(x-3)# increasing or decreasing at #x=2#?
- How do you use the intermediate value theorem to show that there is a root of the equation #e^(-x^2)# over the interval (0,1)?
- What are the extrema of #f(x) = 3x^2 + 12x + 16#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7