How do use the first derivative test to determine the local extrema #y= (x²-3x+3)/ (x-1) #?
There is a local maximum of
and a local minimum of
Differentiate using the quotient rule and simplify to get:
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To use the first derivative test to determine the local extrema of ( y = \frac{x^2 - 3x + 3}{x - 1} ), follow these steps:
- Find the derivative of the function ( y ) with respect to ( x ).
- Set the derivative equal to zero and solve for ( x ) to find critical points.
- Determine the intervals on the number line using the critical points.
- Test the sign of the derivative in each interval to identify where the function is increasing or decreasing.
- Determine the nature of the extrema based on the behavior of the derivative at each critical point.
Let's go through each step:
-
( 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} )
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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.
-
Determine the intervals on the number line using the critical points.
-
Test the sign of the derivative in each interval to identify where the function is increasing or decreasing.
-
Determine the nature of the extrema based on the behavior of the derivative at each critical point.
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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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