How do you use the first and second derivatives to sketch #y = x^3  12x  12#?
See the answer bellow.
Use the information from the first and second derivative to sketch the graph as well as the y intercept.
graph{y=x^312x12 [10, 10, 5, 5]}
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To sketch the graph of ( y = x^3  12x  12 ) using the first and second derivatives, follow these steps:

Find the first derivative ( y' ) by differentiating the function ( y ) with respect to ( x ). [ y' = 3x^2  12 ]

Set ( y' = 0 ) and solve for ( x ) to find critical points. [ 3x^2  12 = 0 ] [ x^2 = 4 ] [ x = \pm 2 ]

Test the intervals between and around the critical points using the first derivative test to determine the increasing and decreasing intervals.

Find the second derivative ( y'' ) by differentiating ( y' ) with respect to ( x ). [ y'' = 6x ]

Determine the concavity of the graph by analyzing the sign of the second derivative in each interval.

Use the critical points, increasing/decreasing intervals, and concavity information to sketch the graph of the function ( y = x^3  12x  12 ).
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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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