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 determine 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. a. the coordinates of the point(s) of inflection are (simplify your answer. type an ordered pair. use a comma to separate answers as needed.) b. there are no points of inflection. determine the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is(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 x - intercepts.

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 determine 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. a. the coordinates of the point(s) of inflection are (simplify your answer. type an ordered pair. use a comma to separate answers as needed.) b. there are no points of inflection. determine the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is(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 x - intercepts.

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 $f^\prime(x)=3x^{-2}=\frac{3}{x^{2}}$. Since $f^\prime(x)=\frac{3}{x^{2}}>0$ for all $x\neq0$, the function is increasing on $(-\infty,0)$ and $(0,\infty)$.

Step3: Find the second - derivative

Differentiate $f^\prime(x)=\frac{3}{x^{2}} = 3x^{-2}$ with respect to $x$. Using the power rule, $f^{\prime\prime}(x)=-6x^{-3}=-\frac{6}{x^{3}}$.

Step4: Determine points of inflection

Set $f^{\prime\prime}(x) = 0$. But $-\frac{6}{x^{3}}=0$ has no solution for real - valued $x$. So there are no points of inflection.

Step5: Find the x - intercept

Set $y = f(x)=0$, so $\frac{8x - 3}{x}=0$. This implies $8x-3 = 0$ (since $x\neq0$ when considering the original rational function) and $x=\frac{3}{8}$. The x - intercept is the point $(\frac{3}{8},0)$.

Answer:

  1. B. There are no points of inflection.
  2. A. The x - intercept(s) is(are) $(\frac{3}{8},0)$