Over what intervals is # f(x)=(9x^2-x)/(x-1) # increasing and decreasing?

Answer 1

#f# is increasing on #(-oo,(3-2sqrt2)/3)# and on #((3+2sqrt2)/3,oo)# and it is decreasing on #((3-2sqrt2)/3,1)# and on #(1,(3+2sqrt2)/3)#

#f'(x) = ((18x-1)(x-1)-(9x^2-x)(1))/(x-1)^2#
# = (9x^2-18x+1)/(x-1)^2#
#f'(x) = 0#, at solutions to #9x^2-18x+1=0#. Solve by completing the square of by using the quadratic formula. #x=(3+-2sqrt2)/3#
#f'(x)# is not defined at #x=0#
Test the sign of #f'# on each interval:

#{: (bb"Interval:",(-oo,(3-2sqrt2)/3),((3-2sqrt2)/3,1),(1,(3+2sqrt2)/3),((3+2sqrt2)/3,oo)), (darrbb"Factors"darr,"========","======","=====","======"), (9x^2-18x+1, bb" +",bb" -",bb" -",bb" +"), ((x-1)^2,bb" +",bb" +",bb" +",bb" +"), ("==========","========","======","=====","======"), (bb"Product"=f'(x),bb" +",bb" -",bb" -",bb" +") :}#

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Answer 2

To determine the intervals where ( f(x) = \frac{9x^2 - x}{x - 1} ) is increasing or decreasing, follow these steps:

  1. Find the derivative of ( f(x) ) with respect to ( x ).
  2. Set the derivative equal to zero and solve for ( x ) to find critical points.
  3. 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:

  1. 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} ]

  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} ]

  3. 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} ).

  1. 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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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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