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

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

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

Answer

Explanation:

Step1: Find the domain

Factor the denominator $x^{2}-8x - 9=(x - 9)(x+1)$. The domain is all real - numbers except $x=-1$ and $x = 9$, i.e., $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$.

Step2: Differentiate the function

Use the quotient rule. If $f(x)=\frac{u}{v}$ where $u=x + 1$ and $v=x^{2}-8x - 9$, then $u'=1$ and $v'=2x-8$. The quotient rule states that $f'(x)=\frac{u'v - uv'}{v^{2}}$. So $f'(x)=\frac{(x^{2}-8x - 9)-(x + 1)(2x - 8)}{(x^{2}-8x - 9)^{2}}=\frac{x^{2}-8x - 9-(2x^{2}-8x+2x - 8)}{(x^{2}-8x - 9)^{2}}=\frac{x^{2}-8x - 9-2x^{2}+6x + 8}{(x^{2}-8x - 9)^{2}}=\frac{-x^{2}-2x - 1}{(x^{2}-8x - 9)^{2}}=\frac{-(x + 1)^{2}}{(x^{2}-8x - 9)^{2}}$.

Step3: Analyze the sign of the derivative

Since $(x + 1)^{2}\geq0$ and $(x^{2}-8x - 9)^{2}>0$ for $x\neq - 1,9$, then $f'(x)\leq0$ for all $x$ in the domain. The function is decreasing on $(-\infty,-1)\cup(-1,9)\cup(9,\infty)$.

Step4: Find relative extrema

Since $f'(x)$ never changes sign (is non - positive and never positive) in the domain, there are no relative extrema.

Answer:

For the increasing and decreasing intervals: The function is decreasing on $(-\infty,-1),(-1,9),(9,\infty)$ and never increasing. For the relative extrema: B. There are no relative extrema.