Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=1+3 e^{-0.2 x} \) Find the intervals where \( f(x) \) is increasing or decreasing. 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. The function is increasing on \( \square \). It is never decreasing. B. The function is increasing on \( \square \). It is decreasing on C. The function is decreasing on \( \square \). It is never increasing. Find the location of any local extrema of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. There is a local maximum at \( x=\square \) and there is a local minimum at \( x=\square . \square \). (Type integers or decimals. Use a comma to separate answers as needed.) B. There is a local minimum at \( x=\square \). There is no local maximum. (Type an integer or a decimal. Use a comma to separate answers as needed.) C. There is a local maximum at \( x=\square \). There is no local minimum. (Type an integer or a decimal. Use a comma to separate answers as needed.) D. There are no local extrema.
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To analyze the function \( f(x) = 1 + 3 e^{-0.2 x} \), we start by noting that as \( x \) increases, \( e^{-0.2 x} \) approaches zero, leading \( f(x) \) to approach 1. The derivative \( f'(x) = -0.6 e^{-0.2 x} \) is always negative for all \( x \) (since the exponential is always positive), which indicates that \( f(x) \) is decreasing on the entire interval \( (0, \infty) \). Therefore, the intervals are: **A. The function is decreasing on \( (0, \infty) \). It is never increasing.** For local extrema, since \( f'(x) \) does not change sign and is never zero, we conclude: **D. There are no local extrema.**
