# How do you find the intervals of increasing and decreasing using the first derivative given #y=x^2-6x+8#?

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To find the intervals of increasing and decreasing for the function ( y = x^2 - 6x + 8 ) using the first derivative, follow these steps:

- Find the first derivative of the function ( y' ).
- Set ( y' ) equal to zero and solve for ( x ) to find critical points.
- Test the intervals between critical points by plugging test points into ( y' ) to determine if the function is increasing or decreasing in those intervals.

Here are the detailed steps:

- The first derivative of ( y = x^2 - 6x + 8 ) is ( y' = 2x - 6 ).
- Set ( y' = 0 ) and solve for ( x ): [ 2x - 6 = 0 ] [ 2x = 6 ] [ x = 3 ]

So, the critical point is ( x = 3 ). 3. Test the intervals:

- Choose a test point less than 3, say ( x = 0 ). [ y'(0) = 2(0) - 6 = -6 ] Since ( y'(0) < 0 ), the function is decreasing on the interval ( (-\infty, 3) ).
- Choose a test point greater than 3, say ( x = 4 ). [ y'(4) = 2(4) - 6 = 2 ] Since ( y'(4) > 0 ), the function is increasing on the interval ( (3, \infty) ).

Therefore, the interval of increasing is ( (3, \infty) ), and the interval of decreasing is ( (-\infty, 3) ).

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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.

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