Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{f}(\mathrm{x})=5+2 e^{-0.2 \mathrm{x}} \). Find the intervals where \( \mathrm{f}(\mathrm{x}) \) is concave upward or downward. 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 concave upward on . It is never concave downward. B. The function is concave downward on It is never concave upward. C. The function is concave upward on Find the location of any inflection points of \( \mathrm{f}(\mathrm{x}) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. There is an inflection point at \( \mathrm{x}= \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no inflection points.
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To analyze the function \( f(x) = 5 + 2e^{-0.2x} \), we first observe that it is an exponential function that approaches a horizontal asymptote at \( f(x) = 5 \) as \( x \) increases, starting from \( f(0) = 7 \). The graph is concave upward for all \( x \) because its second derivative remains positive. Thus, we conclude that the function is concave upward on \( (-\infty, \infty) \) and never concave downward. Moving on to inflection points, we compute the first and second derivatives. The second derivative, \( f''(x) \), is always positive, indicating there are no points where the concavity changes. Therefore, the correct choice is that there are no inflection points. **Final Answers:** 1. The function is concave upward on \( (-\infty, \infty) \). It is never concave downward. 2. B. There are no inflection points.