Over what intervals is # f(x)=(9x^2-x)/(x-1) # increasing and decreasing?
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To determine the intervals where ( f(x) = \frac{9x^2 - x}{x - 1} ) is increasing or decreasing, follow these steps:
- Find the derivative of ( f(x) ) with respect to ( x ).
- Set the derivative equal to zero and solve for ( x ) to find critical points.
- Test the intervals between the critical points and the endpoints of the domain to determine where ( f(x) ) is increasing or decreasing.
Here's how to apply these steps:
-
Find the derivative of ( f(x) ) using the quotient rule: [ f'(x) = \frac{(2)(9x^2 - x)(x - 1) - (9x^2 - x)(1)}{(x - 1)^2} ]
-
Simplify the derivative: [ f'(x) = \frac{(18x^2 - 2x)(x - 1) - (9x^2 - x)}{(x - 1)^2} ] [ f'(x) = \frac{18x^3 - 18x^2 - 2x^2 + 2x - 9x^2 + x}{(x - 1)^2} ] [ f'(x) = \frac{18x^3 - 29x^2 + 3x}{(x - 1)^2} ]
-
Set ( f'(x) ) equal to zero and solve for ( x ) to find critical points: [ 18x^3 - 29x^2 + 3x = 0 ]
There are critical points at ( x = 0 ) and ( x = \frac{29 \pm \sqrt{493}}{36} ).
- Test the intervals:
- Choose values of ( x ) in each interval:
- Test ( x = -1 ), ( x = \frac{29 - \sqrt{493}}{36} ), ( x = \frac{29 + \sqrt{493}}{36} ), and ( x = 2 ).
- Determine the sign of ( f'(x) ) in each interval to determine whether ( f(x) ) is increasing or decreasing.
The intervals where ( f(x) ) is increasing are ( (-\infty, \frac{29 - \sqrt{493}}{36}) ) and ( (\frac{29 + \sqrt{493}}{36}, 1) ), and it is decreasing on ( (\frac{29 - \sqrt{493}}{36}, \frac{29 + \sqrt{493}}{36}) ) and ( (1, \infty) ).
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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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