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

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.\n( f(x)=x^{22}+3 x^{2} )\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. ( (-infty, infty) )\n(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.\n(type your answer in interval notation. type an exact answer use a comma to separate answers as needed.)\nthe graph is never concave downward.\ndetermine the ( x ) coordinates of any inflection points of the graph of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na.\n(type an exact answer. use a comma to separate answers as needed)\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.\n( f(x)=x^{22}+3 x^{2} )\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. ( (-infty, infty) )\n(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.\n(type your answer in interval notation. type an exact answer use a comma to separate answers as needed.)\nthe graph is never concave downward.\ndetermine the ( x ) coordinates of any inflection points of the graph of ( f(x) ). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na.\n(type an exact answer. use a comma to separate answers as needed)\nb. there are no inflection points

Answer

Explanation:

Step1: Find the second - derivative

First, find the first - derivative using the power rule (y = x^n), (y^\prime=nx^{n - 1}). For (f(x)=x^{22}+3x^{2}), (f^\prime(x)=22x^{21}+6x). Then find the second - derivative: (f^{\prime\prime}(x)=462x^{20}+6).

Step2: Analyze the sign of the second - derivative

Since (x^{20}\geq0) for all real (x), then (462x^{20}\geq0) for all real (x). So (f^{\prime\prime}(x)=462x^{20}+6>0) for all (x\in(-\infty,\infty)).

Answer:

  • Concave upward: ((-\infty,\infty))
  • Concave downward: The graph is never concave downward.
  • Inflection points: There are no inflection points.