Summarize the pertinent information oblained by applying the graphing strategy and skelch the graph of \( y=I(x) \). \[ 1(x)=\ln (x+8)-1 \] Select the correct choice below and, il necessary, fil in the answer box to complete your choice. A. The function \( f \) is decreasing on the subinterval(s) \( \square \) . (Type your answer in interval notation. Use a comma to separale answers as needed) B. The function 1 is never decreasing Sclect the correct choice below and, il necessary, fill in the answer box to complele your choice. A. The function \( f \) has a local maximum at \( x= \) \( \square \) (Type an exact answer. Use a comma to separate answers as needed.) B. The function \( f \) has no local maximum. Select the correct choice below and, it necessary, fill in the answer box to complele your choice A. Ihe function t has a local minimum at \( x= \) \( \square \) . (Type an exact airswer. Use a comma to separale answers as meeded.) B. The function i has no local minimum.

1. The function is given by \[ I(x)=\ln(x+8)-1. \] **Domain:** Since the argument of the logarithm must be positive, we require \[ x+8>0 \quad \Longrightarrow \quad x>-8. \] Thus, the domain is \(( -8,\infty )\). 2. **Graphing information and monotonicity:** To determine monotonicity, we differentiate the function: \[ I'(x)=\frac{d}{dx}\left[ \ln(x+8)-1 \right]=\frac{1}{x+8}. \] For \(x>-8\), we have \(x+8>0\) so that \[ I'(x)>0. \] This implies that the function is strictly increasing on its entire domain. **Answer for decreasing:** Choice B: The function \( I \) is never decreasing. 3. **Local Maximum:** A strictly increasing function has no turning points where it changes from increasing to decreasing. Therefore, there is no local maximum in the domain. **Answer for local maximum:** Choice B: The function \( I \) has no local maximum. 4. **Local Minimum:** Similarly, since the function is strictly increasing, it does not have an interior point at which it is lower than all nearby points; the lowest values occur as \( x\to -8^+ \), but \( x=-8 \) is not in the domain. Therefore, there is no local minimum. **Answer for local minimum:** Choice B: The function \( I \) has no local minimum. 5. **Graph Sketch:** - The graph has a vertical asymptote at \( x=-8 \) since \(\ln(x+8)\) tends to \(-\infty\) as \(x\to -8^+\). - As \( x \to -8^+ \), \( I(x)\to -\infty \). - The function is strictly increasing, passing through the point when \(x=-7\) or any other sample point; for example, for \( x=0 \): \[ I(0)=\ln(0+8)-1=\ln8-1. \] - As \( x\to\infty \), \( I(x)\to\infty \). Thus, the final answers are: - Decreasing: B (The function is never decreasing) - Local Maximum: B (The function has no local maximum) - Local Minimum: B (The function has no local minimum)