How do you use the first and second derivatives to sketch #y= (x2) (x+2) (x4)#?
graph{(x2)(x+2)(x4) [10, 10, 20, 20]}
We now have enough to sketch the curve: graph{(x2)(x+2)(x4) [10, 10, 20, 20]}
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To sketch the graph of ( y = (x2)(x+2)(x4) ) using the first and second derivatives, follow these steps:

Find the First Derivative: Calculate the first derivative ( y' ) to identify critical points (where ( y' = 0 )) and determine the intervals of increase and decrease.
[ y' = 1(x+2)(x4)  (x2)(x4)  (x2)(x+2) ] Simplify to: [ y' = (x^2  2x  8)  (x^2  6x + 8)  (x^2  4) ] [ y' = x^2 + 2x + 8  x^2 + 6x  8  x^2 + 4 ] [ y' = 3x^2 + 8x  4 ]

Find Critical Points: Solve for ( x ) when ( y' = 0 ).
[ 3x^2 + 8x  4 = 0 ] Factor or use the quadratic formula to find the roots.

Determine Intervals of Increase and Decrease: Use test points from each interval to determine if the function is increasing or decreasing.

Find the Second Derivative: Calculate the second derivative ( y'' ) to identify points of inflection and concavity.
[ y'' = \frac{d}{dx}(3x^2 + 8x  4) ] [ y'' = 6x + 8 ]

Find Points of Inflection: Solve for ( x ) when ( y'' = 0 ) to find potential points of inflection.

Determine Concavity: Use test points from each interval to determine if the function is concave up or concave down.

Sketch the Graph:
 Plot the critical points, points of inflection, and any additional points determined from your tests.
 Use the information from the intervals of increase/decrease and concavity to sketch the curve.
 Ensure the curve passes through any intercepts or asymptotes.
Remember, this process provides a general sketch of the function. For more accurate details, plotting points or using graphing software can be beneficial.
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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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