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}+3x^{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 (-∞,∞)
(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.
a.
(type your answer in interval notation type an exact answer use a comma to separate answers as needed)
b. the graph is never concave downward

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.