Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)= \) In \( \left(x^{2}+9\right) \). A. \( f(x) \) is increasing on \( (0, \infty) \) and decreasing on \( (-\infty, 0) \). B. \( f(x) \) is increasing on C. \( f(x) \) is decreasing on \( f(x) \) has a local Summarize the (Type your ans A. \( f(x) \) is cc minimed by analyzing \( f^{\prime \prime}(x) \) Select the correct choice below and fill in the answer box(es) to complete your choice. B. \( f(x) \) is cc maximum. C. \( f(x) \) is concave downward on to separate answers as needed.)

The function \( f(x) = \ln(x^2 + 9) \) is defined for all \( x \) in the real numbers since \( x^2 + 9 \) is always positive. By applying the graphing strategy, we find that \( f(x) \) is an increasing function on \( (0, \infty) \) and exhibits a decreasing behavior on \( (-\infty, 0) \). Notably, the function doesn’t have a local maximum or minimum; instead, it approaches \(\ln(9)\) as \( x \) approaches 0 from either side. For concavity, we analyze the second derivative \( f''(x) \). After performing the necessary calculations, we find that \( f(x) \) is concave downward on \( (-\infty, \infty) \). Hence, the choices can be summarized as follows: A. \( f(x) \) is increasing on \( (0, \infty) \) and decreasing on \( (-\infty, 0) \). B. \( f(x) \) is concave downward on \( (-\infty, \infty) \).