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, and where any intercepts occur.\n f(x)=\frac{5}{x + 7} \ndetermine the slant asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the slant asymptote(s) is(are) \n(type an equation. use a comma to separate answers as needed.)\nb. the 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 up on and concave down on \n(simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nb. 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.)\nc. 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.)

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, and where any intercepts occur.\n f(x)=\frac{5}{x + 7} \ndetermine the slant asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the slant asymptote(s) is(are) \n(type an equation. use a comma to separate answers as needed.)\nb. the 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 up on and concave down on \n(simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nb. 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.)\nc. 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.)

Answer

Explanation:

Step1: Find domain

The function $f(x)=\frac{5}{x + 7}$ is undefined when $x+7=0$, so the domain is $x\neq - 7$.

Step2: Find vertical asymptote

Set the denominator equal to zero. $x+7 = 0$ gives $x=-7$, so the vertical asymptote is $x=-7$.

Step3: Find horizontal asymptote

Since the degree of the numerator is 0 and the degree of the denominator is 1, and $0<1$, the horizontal asymptote is $y = 0$. There are no slant asymptotes (because the degree of the numerator is less than the degree of the denominator). So the answer for slant - asymptote is:

Answer:

B. The function has no slant asymptotes

Step4: Find first - derivative

Using the quotient rule, if $y=\frac{u}{v}$ where $u = 5$ and $v=x + 7$, then $y^\prime=\frac{u^\prime v-uv^\prime}{v^{2}}$. Here, $u^\prime=0$ and $v^\prime = 1$. So $y^\prime=-\frac{5}{(x + 7)^{2}}$. Since $y^\prime<0$ for all $x\neq - 7$, the function is decreasing on $(-\infty,-7)\cup(-7,\infty)$.

Step5: Find second - derivative

Using the quotient rule again on $y^\prime=-\frac{5}{(x + 7)^{2}}=-5(x + 7)^{-2}$. Let $u=-5$ and $v=(x + 7)^{2}$, then $y^{\prime\prime}=\frac{10}{(x + 7)^{3}}$.

Step6: Find concavity

Set $y^{\prime\prime}=0$, but $\frac{10}{(x + 7)^{3}}=0$ has no solution. When $x<-7$, $y^{\prime\prime}<0$, so the function is concave down on $(-\infty,-7)$. When $x>-7$, $y^{\prime\prime}>0$, so the function is concave up on $(-7,\infty)$.

Answer:

A. The function is concave up on $(-7,\infty)$ and concave down on $(-\infty,-7)$