Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=e^{0.5 x}+324 e^{-0.5 x} \) (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The function \( f \) is never concave upward. A. The function \( f \) is concave downward on the subinterval(s) (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The function \( f \) is never concave downward. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) has an inflection point at \( x= \) B. The function in the answer box to complete your choice. (Type an exact answer. Use a comma to separate answers as needed.) Blection point.
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To analyze the function \( f(x) = e^{0.5 x} + 324 e^{-0.5 x} \), first, we can determine where its concavity changes. By calculating the first and second derivatives, we find that the second derivative is always negative, indicating that the function is concave downward for all \( x \). Therefore, the function is concave downward on the interval \( (-\infty, \infty) \). The function \( f \) does not have any inflection points as it is never concave upward. So, there is no \( x \) value where the concavity changes. The summary of our findings is: The function \( f \) is concave downward on the interval \((- \infty, \infty)\), has no inflection points, and is never concave upward.
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