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]}
By signing up, you agree to our Terms of Service and Privacy Policy
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.
By signing up, you agree to our Terms of Service and Privacy Policy
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.
 How do you use the second derivative test to find min/max/pt of inflection of #y= x^55x#?
 How do you find the inflection points for the function #f(x)=x/(x1)#?
 Is #f(x)=x^52x^26x+3# concave or convex at #x=4#?
 How do you find all points of inflection given #y=2x^2+4x+4#?
 For what values of x is #f(x)=(x1)(x6)(x2)# concave or convex?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7