Question

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, where any points of inflection occur, and where any intercepts occur.
f(x)=\frac{5}{x + 7}
on 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.
a. the function is increasing on and decreasing on
(simplify your answers. type your answers in interval notation. type exact answers, using radicals as needed. use a comma to separate answers as needed.)
b. the function is decreasing on the function is never increasing.
(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.)
c. the function is increasing on the function is never decreasing.
(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.)
d. the function is never increasing or decreasing.
determine the coordinates of the relative extrema. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the coordinates of the relative extrema are
(type an ordered - pair. type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. there are no relative extrema.

Explanation:

Step1: Find the derivative

First, rewrite the function as $f(x)=5(x + 7)^{-1}$. Using the power - rule for differentiation $(u^n)'=nu^{n - 1}u'$, where $u=x + 7$ and $n=-1$. The derivative $f'(x)=5\times(-1)\times(x + 7)^{-2}\times1=-\frac{5}{(x + 7)^2}$.

Step2: Analyze increasing and decreasing intervals

Since $f'(x)=-\frac{5}{(x + 7)^2}<0$ for all $x
eq - 7$, the function is decreasing on $(-\infty,-7)\cup(-7,\infty)$ and never increasing.

Step3: Analyze relative extrema

A relative extremum occurs where $f'(x) = 0$ or $f'(x)$ is undefined and the sign of $f'(x)$ changes. Since $f'(x)=-\frac{5}{(x + 7)^2}
eq0$ for all real $x$ (and is undefined at $x=-7$ but the sign of $f'(x)$ does not change around $x =-7$), there are no relative extrema.

Answer:

For the increasing - decreasing question: B. The function is decreasing on $(-\infty,-7)\cup(-7,\infty)$. The function is never increasing.
For the relative extrema question: B. There are no relative extrema.