graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the…

graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the derivatives ( y ^ { prime } ) and ( y ^ { prime prime } ), finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any. (type your answer in interval notation. use a comma to separate answers as needed.) c. the curve increases on the open interval(s) and does not decrease. (type your answer in interval notation. use a comma to separate answers as needed.) d. the curve neither increases nor decreases. identify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection point(s) is/are (type an ordered pair. use a comma to separate answers as needed.) b. there are no inflection points

graph the function ( y = x ^ { 2 } - 2 x - 8 ) by identifying the domain and any symmetries, finding the derivatives ( y ^ { prime } ) and ( y ^ { prime prime } ), finding the critical points and identifying the functions behavior at each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. then find coordinates of absolute extreme points, if any. (type your answer in interval notation. use a comma to separate answers as needed.) c. the curve increases on the open interval(s) and does not decrease. (type your answer in interval notation. use a comma to separate answers as needed.) d. the curve neither increases nor decreases. identify any inflection points. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection point(s) is/are (type an ordered pair. use a comma to separate answers as needed.) b. there are no inflection points

Answer

Explanation:

Step1: Find the first derivative

The function is (y = x^{2}-2x - 8). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (y^\prime=\frac{d}{dx}(x^{2}-2x - 8)=2x-2).

Step2: Find the critical points

Set (y^\prime = 0), so (2x-2 = 0). Solving for (x): (2x=2), then (x = 1).

Step3: Determine the intervals of increase and decrease

We use a test - point method. For the interval ((-\infty,1)), let (x = 0). Then (y^\prime(0)=2\times0 - 2=-2<0), so the function is decreasing on ((-\infty,1)). For the interval ((1,\infty)), let (x = 2). Then (y^\prime(2)=2\times2 - 2 = 2>0), so the function is increasing on ((1,\infty)).

Step4: Find the second derivative

Differentiate (y^\prime=2x - 2) with respect to (x). Using the power rule, (y^{\prime\prime}=\frac{d}{dx}(2x - 2)=2).

Step5: Analyze concavity and inflection points

Since (y^{\prime\prime}=2>0) for all (x\in R), the function is concave up on ((-\infty,\infty)). An inflection point occurs where (y^{\prime\prime}=0) or (y^{\prime\prime}) is undefined. Since (y^{\prime\prime}=2\neq0) for all (x) and (y^{\prime\prime}) is a constant (defined for all (x)), there are no inflection points.

Answer:

For the increasing/decreasing part: The curve increases on the open interval ((1,\infty)) and decreases on the open interval ((-\infty,1)). For the inflection - point part: B. There are no inflection points.