Over what intervals is # f(x)=(9x^2x)/(x1) # 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 onesided 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 onesided 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 onesided 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 onesided 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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