How do use the first derivative test to determine the local extrema #y= (x²-3x+3)/ (x-1) #?

Answer 1

There is a local maximum of #-3# at #x=0#
and a local minimum of #1# at #x=2#.

Let #f(x) = (x²-3x+3)/(x-1)#

Differentiate using the quotient rule and simplify to get:

#f'(x) = (x^2-2x)/(x-1)^2 = (x(x-2))/(x-1)^2#
#f'(x) = 0# at #0# and #2# and #f'(x)# is not defined at #1#.
The domain of #f# includes all Real numbers except #1#,
so the critical numbers for #f# are: #0# and #2#
Test the critical number #0#
If #x# is a little less that #0#, then #x# and #x-2# are negative, while #(x-1)^2# is positive, so #f'(x) > 0#
When #x# is a little greater than #0#, the sign of #x# is positive, but the signs of the other factor remain the same. So #f'(x)# changes to a negative value.
This tells us that #f(0)# (which is #-3#) is a local maximum.
Test the critical number #2#
For#x# a little less than #2#, #f'(x) < 0# and For #x# a little greater than #2#, the sign changes to #f'(x) > 0#
Therefore #f(2)# (which is #(4-6+3)/1 = 1#) is a local minimum.
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Answer 2

To use the first derivative test to determine the local extrema of ( y = \frac{x^2 - 3x + 3}{x - 1} ), follow these steps:

  1. Find the derivative of the function ( y ) with respect to ( x ).
  2. Set the derivative equal to zero and solve for ( x ) to find critical points.
  3. Determine the intervals on the number line using the critical points.
  4. Test the sign of the derivative in each interval to identify where the function is increasing or decreasing.
  5. Determine the nature of the extrema based on the behavior of the derivative at each critical point.

Let's go through each step:

  1. ( y' = \frac{d}{dx}\left(\frac{x^2 - 3x + 3}{x - 1}\right) ) Using the quotient rule, the derivative is: ( y' = \frac{(x - 1)(2x - 3) - (x^2 - 3x + 3)(1)}{(x - 1)^2} )

  2. Set the derivative equal to zero and solve for ( x ): ( \frac{(x - 1)(2x - 3) - (x^2 - 3x + 3)(1)}{(x - 1)^2} = 0 ) Solve for ( x ) to find critical points.

  3. Determine the intervals on the number line using the critical points.

  4. Test the sign of the derivative in each interval to identify where the function is increasing or decreasing.

  5. Determine the nature of the extrema based on the behavior of the derivative at each critical point.

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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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