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.\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

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.\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

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (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). So (f^{\prime\prime}(x)=462x^{20}+6>0) for all (x\in(-\infty,\infty)).

Answer:

A. ((-\infty,\infty))