find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is…

find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points.\nf(x)=x³ - 18x² + 19x + 3\nfor what interval(s) of x is the graph of f concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave upward.\nfor what interval(s) of x is the graph of f concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave downward.\ndetermine the x, y coordinates of the inflection point of the graph of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type an ordered pair.)\nb. there are no inflection points.

find the intervals on which the graph of f is concave upward, the intervals on which the graph of f is concave downward, and the inflection points.\nf(x)=x³ - 18x² + 19x + 3\nfor what interval(s) of x is the graph of f concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave upward.\nfor what interval(s) of x is the graph of f concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave downward.\ndetermine the x, y coordinates of the inflection point of the graph of f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. (type an ordered pair.)\nb. there are no inflection points.

Answer

Explanation:

Step1: Find the first and second derivatives

Given ( f(x)=x^{3}-18x^{2}+19x + 3). The first - derivative (f^{\prime}(x)) using the power rule ((x^{n})^\prime=nx^{n - 1}) is: (f^{\prime}(x)=3x^{2}-36x + 19). The second - derivative (f^{\prime\prime}(x)) is: (f^{\prime\prime}(x)=6x-36).

Step2: Find the critical points of (f^{\prime\prime}(x))

Set (f^{\prime\prime}(x) = 0). (6x-36=0). Add 36 to both sides: (6x=36). Divide both sides by 6: (x = 6).

Step3: Test the intervals for concavity

We have two intervals to test: ((-\infty,6)) and ((6,\infty)).

  • For the interval ((-\infty,6)), let's choose (x = 0). Then (f^{\prime\prime}(0)=6\times0 - 36=-36<0). So the function (y = f(x)) is concave downward on the interval ((-\infty,6)).
  • For the interval ((6,\infty)), let's choose (x = 7). Then (f^{\prime\prime}(7)=6\times7-36 = 6>0). So the function (y = f(x)) is concave upward on the interval ((6,\infty)).

Step4: Find the inflection point

Substitute (x = 6) into (f(x)): (f(6)=6^{3}-18\times6^{2}+19\times6 + 3=216-648+114 + 3=-315).

Answer:

  • For the interval of (x) where the graph of (f) is concave upward: ((6,\infty)).
  • For the interval of (x) where the graph of (f) is concave downward: ((-\infty,6)).
  • The inflection point is ((6,-315)).