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, where any points of inflection occur, and where any intercepts occur. f(x) = (8x - 3)/x b. 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.) c. 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.) 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.

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) = (8x - 3)/x b. 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.) c. 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.) 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.

Answer

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{8x - 3}{x}=8-\frac{3}{x}$.

Step2: Find the first - derivative

Using the power rule, if $y = 8-3x^{-1}$, then $f^\prime(x)=3x^{-2}=\frac{3}{x^{2}}$.

Step3: Analyze increasing and decreasing intervals

Since $f^\prime(x)=\frac{3}{x^{2}}>0$ for all $x\neq0$, the function is increasing on $(-\infty,0)\cup(0,\infty)$ and never decreasing.

Step4: Find the second - derivative

$f^{\prime\prime}(x)=-6x^{-3}=-\frac{6}{x^{3}}$.

Step5: Analyze concavity and inflection points

Set $f^{\prime\prime}(x) = 0$, but $-\frac{6}{x^{3}}=0$ has no solution. When $x<0$, $f^{\prime\prime}(x)>0$, the function is concave up on $(-\infty,0)$; when $x > 0$, $f^{\prime\prime}(x)<0$, the function is concave down on $(0,\infty)$. There are no inflection points as the function is not continuous at $x = 0$.

Step6: Find asymptotes

Vertical asymptote: Set the denominator of the original function equal to 0, $x = 0$ is a vertical asymptote. Horizontal asymptote: $\lim_{x\rightarrow\pm\infty}(8-\frac{3}{x})=8$, so $y = 8$ is a horizontal asymptote.

Step7: Find intercepts

$x$-intercept: Set $y = 0$, then $8-\frac{3}{x}=0$, $8x-3=0$, $x=\frac{3}{8}$. $y$-intercept: The function is not defined at $x = 0$, so there is no $y$-intercept.

Step8: Analyze relative extrema

Since $f^\prime(x)$ never changes sign, there are no relative extrema.

Answer:

The function is increasing on $(-\infty,0)\cup(0,\infty)$ and decreasing on none. There are no relative extrema.