Question

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.
( f(x)=x^{22}+3 x^{2} )
for 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.
a. ( (-infty, infty) )
(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)
b. the graph is never concave upward.
for 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.
(type your answer in interval notation. type an exact answer use a comma to separate answers as needed.)
the graph is never concave downward.
determine 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.
a.
(type an exact answer. use a comma to separate answers as needed)
b. there are no inflection points

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.