Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)= \) In \( \left(x^{2}+9\right) \). Summarize the pertinent information obtained by analyzing \( f^{\prime}(x) \). Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. \( f(x) \) is increasing on \( \square \) and decreasing on \( \square \) B. \( f(x) \) is increasing on C. \( f(x) \) is decreasing on \( f(x) \) has a local Summarize the pertinent information obtained by analyzing \( f^{\prime \prime}(x) \). Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. \( f(x) \) is concave upward on \( \square \) and concave downward on B. \( f(x) \) is concave upward on \( \square \). C. \( f(x) \) is concave downward on \( \square \).

To analyze the function \( f(x) = \ln(x^{2} + 9) \), we start by finding the first derivative \( f'(x) \) to determine the intervals where the function is increasing or decreasing. The derivative is \( f'(x) = \frac{2x}{x^{2} + 9} \). Setting this equal to zero shows that \( f'(x) = 0 \) at \( x = 0 \). Since the function remains positive for \( x > 0 \) and negative for \( x < 0 \), we conclude that \( f(x) \) is increasing on \( (-\infty, 0) \) and \( (0, \infty) \), and decreasing on \( \emptyset \) (as it does not decrease in any interval). Next, we analyze the second derivative \( f''(x) \) to determine the concavity of the function. After computing \( f''(x) \) and analyzing its sign, we find that it is positive when \( x \neq 0 \). Thus, \( f(x) \) is concave upward on \( (-\infty, \infty) \) and never concave downward. So, for your choices: A. \( f(x) \) is increasing on \( (-\infty, 0) \) and \( (0, \infty) \) and decreasing on \( \emptyset \). A. \( f(x) \) is concave upward on \( (-\infty, \infty) \) and concave downward on \( \emptyset \).