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.\n f(x)=\frac{5}{x + 7} \ndetermine 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.\na. the coordinates of the point(s) of inflection are \n(simplify your answer. type an ordered - pair. use a comma to separate answers as needed.)\nb. there are no points of inflection\ndetermine the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the x - intercept(s) is(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 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.\n f(x)=\frac{5}{x + 7} \ndetermine 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.\na. the coordinates of the point(s) of inflection are \n(simplify your answer. type an ordered - pair. use a comma to separate answers as needed.)\nb. there are no points of inflection\ndetermine the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the x - intercept(s) is(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 x - intercepts

Answer

Explanation:

Step1: Recall the formula for the second - derivative

First, rewrite the function (y = f(x)=\frac{5}{x + 7}=5(x + 7)^{-1}). Using the power - rule ((x^n)^\prime=nx^{n - 1}), the first - derivative (y^\prime=f^\prime(x)=-5(x + 7)^{-2}). Then, the second - derivative (y^{\prime\prime}=f^{\prime\prime}(x)=10(x + 7)^{-3}=\frac{10}{(x + 7)^3}).

Step2: Find points of inflection

Points of inflection occur where (y^{\prime\prime}=0) or (y^{\prime\prime}) is undefined and the concavity changes. Set (y^{\prime\prime}=0), (\frac{10}{(x + 7)^3}=0) has no solution since the numerator is non - zero. The second - derivative is undefined at (x=-7), but the function (y = f(x)) is not continuous at (x=-7) (it has a vertical asymptote at (x=-7)). So, there are no points of inflection.

Step3: Find x - intercepts

The x - intercepts occur where (y = 0). Set (y=\frac{5}{x + 7}=0). Since the numerator (5\neq0), the equation (\frac{5}{x + 7}=0) has no solution. So, there are no x - intercepts.

Answer:

Determine the coordinates of the point(s) of inflection: B. There are no points of inflection Determine the x - intercept(s): B. There are no x - intercepts