Summarize the pertinent information ublained by applying the graphing slrategy and sketch the graph of \( y=f(x) \). \[ f(x)=\ln (x+8)-1 \] Select the correct choice below and, il necessary, fill in the answer box to complete your choice. A. The function f is concave upward on the subinterval( s ) \( \square \) 1. (Type your answer in interval nolation. Type an oxacl answer. Use a comma to separate answers as noeded.) B. The function \( f \) is nover concave upward. Select the correct choice below and, if necessary, fill in the answer box to complele your chorce. A. Tho function \( f \) is concavo downward on the subinleval(s) \( (-8, \infty) \). (Type your answer in interval notation. Type an exact answer. Use a comma lo separate answers as needed.) B. The function \( f \) is never concave downward. Select the correct choive below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) has an inflection point al \( x= \) \( \square \) . (Type an exact answer. Use a comma lo separate answers as needed) B. The function I has no inflection point.

**Step 1. Determine the Domain and Basic Shape** The function is \[ f(x)=\ln(x+8)-1. \] Since the logarithm is only defined when its argument is positive, we require \[ x+8>0 \quad \Rightarrow \quad x>-8. \] Thus, the domain is \((-8,\infty)\). The graph is that of the logarithm function shifted 8 units to the left and 1 unit downward. **Step 2. Find the First and Second Derivatives** The first derivative is \[ f'(x)=\frac{1}{x+8}, \] which shows that the function is increasing on its domain (since \(x+8>0\)). The second derivative is \[ f''(x)=-\frac{1}{(x+8)^2}. \] Since \((x+8)^2\) is always positive for \(x>-8\), we have \[ f''(x)<0 \quad \text{for all } x>-8. \] **Step 3. Analyze Concavity** Because \(f''(x)<0\) throughout the domain, the function is: - Never concave upward. - Concave downward on the entire interval \((-8,\infty)\). **Step 4. Identify Inflection Points** An inflection point occurs where the concavity changes (i.e., where \(f''(x)=0\) or is undefined and changes sign). Here, \(f''(x)\) is never zero and does not change sign; therefore, the function has no inflection points. **Final Answers:** - **On concavity upward:** **Choice B:** The function \(f\) is never concave upward. - **On concavity downward:** **Choice A:** The function \(f\) is concave downward on the interval \((-8,\infty)\). - **On inflection points:** **Choice B:** The function \(f\) has no inflection point.