graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y…

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, 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. identify any local maxima. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/maxima is/are located at (type an ordered pair. use a comma to separate answers as needed.) b. there are no local maxima. identify where the curve is increasing or decreasing. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve does not increase and decreases on the open interval(s) (type your answer in interval notation. use a comma to separate answers as needed.) b. 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.) c. the curve increases on the open interval(s) and decreases on the open interval(s)

graph the function y = x² - 2x - 8 by identifying the domain and any symmetries, finding the derivatives y and y, 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. identify any local maxima. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/maxima is/are located at (type an ordered pair. use a comma to separate answers as needed.) b. there are no local maxima. identify where the curve is increasing or decreasing. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve does not increase and decreases on the open interval(s) (type your answer in interval notation. use a comma to separate answers as needed.) b. 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.) c. the curve increases on the open interval(s) and decreases on the open interval(s)

Answer

Explanation:

Step1: Find the first - derivative

Given $y = x^{2}-2x - 8$, using the power rule $(x^n)'=nx^{n - 1}$, we have $y'=2x-2$.

Step2: Find the critical points

Set $y' = 0$, so $2x-2=0$. Solving for $x$ gives $x = 1$.

Step3: Analyze the sign of the first - derivative

Choose a test point in the interval $(-\infty,1)$, say $x = 0$. Then $y'(0)=2\times0 - 2=-2<0$, so the function is decreasing on $(-\infty,1)$. Choose a test point in the interval $(1,\infty)$, say $x = 2$. Then $y'(2)=2\times2 - 2 = 2>0$, so the function is increasing on $(1,\infty)$.

Step4: Determine local maxima/minima

Since the function changes from decreasing to increasing at $x = 1$, there is a local minimum at $x = 1$. Substitute $x = 1$ into the original function: $y(1)=1^{2}-2\times1 - 8=-9$. So there are no local maxima.

Step5: Find the second - derivative

Differentiate $y'=2x - 2$ with respect to $x$ to get $y'' = 2>0$. The function is concave up everywhere.

Answer:

For local maxima: B. There are no local maxima. For increasing and decreasing: C. The curve increases on the open interval $(1,\infty)$ and decreases on the open interval $(-\infty,1)$