sketch the graph of the following function. indicate where the function is increasing or decreasing, where…

sketch the graph of the following function. indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down.\nf(x) = \\frac{x + 1}{x^{2}-8x - 9}\n(type an equation. use a comma to separate answers as needed.)\nthe function has no slant asymptotes.\non what interval(s) is f concave up and on what interval(s) is f concave down? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is concave down on and is never concave up.\n(simplify your answer. type your answer in interval - notation. use a comma to separate answers as needed.)\nb. the function is concave up on (0,\\infty) and concave down on (-\\infty,-1),(-1,0).\n(simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is concave up on and is never concave down.\n(simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\ndetermine the coordinates of the point(s) of inflection. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the coordinates of the point(s) of inflection are \n(simplify your answer. type an ordered - pair. use a comma to separate answers as needed.)\nb. there are no points of inflection.

sketch the graph of the following function. indicate where the function is increasing or decreasing, where any relative extrema occur, where asymptotes occur, where the graph is concave up or concave down.\nf(x) = \\frac{x + 1}{x^{2}-8x - 9}\n(type an equation. use a comma to separate answers as needed.)\nthe function has no slant asymptotes.\non what interval(s) is f concave up and on what interval(s) is f concave down? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is concave down on and is never concave up.\n(simplify your answer. type your answer in interval - notation. use a comma to separate answers as needed.)\nb. the function is concave up on (0,\\infty) and concave down on (-\\infty,-1),(-1,0).\n(simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nc. the function is concave up on and is never concave down.\n(simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\ndetermine the coordinates of the point(s) of inflection. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the coordinates of the point(s) of inflection are \n(simplify your answer. type an ordered - pair. use a comma to separate answers as needed.)\nb. there are no points of inflection.

Answer

Explanation:

Step1: First, factor the denominator

Given $f(x)=\frac{x + 1}{x^{2}-8x - 9}=\frac{x + 1}{(x - 9)(x+1)}$ (for $x\neq - 1$), which simplifies to $f(x)=\frac{1}{x - 9}$ for $x\neq - 1$.

Step2: Find the first - derivative

Using the quotient rule, if $y=\frac{u}{v}$ where $u = 1$ and $v=x - 9$, then $y^\prime=-\frac{1}{(x - 9)^{2}}$.

Step3: Find the second - derivative

Differentiating $y^\prime=-\frac{1}{(x - 9)^{2}}=-(x - 9)^{-2}$ using the power rule, we get $y^{\prime\prime}=\frac{2}{(x - 9)^{3}}$.

Step4: Determine concavity

Set $y^{\prime\prime}=0$, but $\frac{2}{(x - 9)^{3}}=0$ has no solution. We consider the sign of $y^{\prime\prime}$ for different intervals. When $x>9$, $y^{\prime\prime}>0$, so the function is concave up on the interval $(9,\infty)$. When $x<9$, $y^{\prime\prime}<0$, so the function is concave down on the interval $(-\infty,9)$. Since the original function is undefined at $x=-1$ and $x = 9$, we need to be careful. But overall, the function $y = f(x)$ is concave up on $(9,\infty)$ and concave down on $(-\infty,-1),(-1,9)$.

Step5: Find points of inflection

Since $y^{\prime\prime}$ never changes sign across a real - valued $x$ value (it is undefined at $x = 9$ but does not change sign there in the domain of the original non - simplified function considering the removable discontinuity at $x=-1$), there are no points of inflection.

Answer:

The function is concave up on $(9,\infty)$ and concave down on $(-\infty,-1),(-1,9)$. There are no points of inflection.