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 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. identify where the curve is concave up or concave down. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve is concave up on the interval(s) and is concave down on the interval(s). (type your answers in interval notation. use a comma to separate answers as needed.) b. the curve is concave up on the interval(s) and is never concave down. (type your answer in interval notation. use a comma to separate answers as needed.) c. the curve is never concave up and is concave down on the 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 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. identify where the curve is concave up or concave down. select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the curve is concave up on the interval(s) and is concave down on the interval(s). (type your answers in interval notation. use a comma to separate answers as needed.) b. the curve is concave up on the interval(s) and is never concave down. (type your answer in interval notation. use a comma to separate answers as needed.) c. the curve is never concave up and is concave down on the interval(s).

Answer

Explanation:

Step1: Find the first - derivative

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

Step2: Find the second - derivative

Differentiate $y'$ with respect to $x$. $y''=\frac{d}{dx}(2x - 2)=2$.

Step3: Find inflection points

Inflection points occur where $y'' = 0$ or $y''$ is undefined. Since $y''=2\neq0$ and is defined for all real $x$, there are no inflection points.

Step4: Determine concavity

Since $y'' = 2>0$ for all $x\in(-\infty,\infty)$, the curve is concave up on the interval $(-\infty,\infty)$ and is never concave down.

Answer:

For the inflection - point question: B. There are no inflection points. For the concavity question: B. The curve is concave up on the interval $(-\infty,\infty)$ and is never concave down.