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)=x^{22}+3x^{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 (-∞,∞)\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.\na.\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 downward

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)=x^{22}+3x^{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 (-∞,∞)\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.\na.\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 downward

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (y = f(x)=x^{22}+3x^{2}), we have (f^\prime(x)=22x^{21}+6x).

Step2: Find the second derivative

Differentiate (f^\prime(x)) again. (f^{\prime\prime}(x)=22\times21x^{20}+6=462x^{20}+6).

Step3: 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:

  • For the concavity upward: A. ((-\infty,\infty))
  • For the concavity downward: B. The graph is never concave downward.