Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( \mathrm{y}=\mathrm{f}(\mathrm{x}) \). \( \mathrm{f}(\mathrm{x})=e^{0.5 \mathrm{x}}+324 e^{-0.5 \mathrm{x}} \) A. The function f has a local maximum at \( \mathrm{x}= \) (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.). B. The function f has no local maximum. A. The function f has a local minimum at \( \mathrm{x}= \) In (324). (Type an exact answer. Use a comma to separate answers as needed.) B. The function f has no local minimum. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) is concave upward 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 upward.
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To analyze \( f(x) = e^{0.5x} + 324 e^{-0.5x} \), we first find its critical points by setting the derivative \( f'(x) \) to zero. Calculating, we find that \( f'(x) \) changes signs around \( x = 0 \), indicating a local minimum there, with \( f(0) = 325 \). Thus, the local minimum occurs at \( x = 0 \) and the minimum value is 325. Now, for concavity, we compute the second derivative \( f''(x) \). The signs of \( f''(x) \) are positive for all \( x \), suggesting that the function is concave upward everywhere. Therefore, the intervals are \( (-\infty, \infty) \). In summary, the function has a local minimum at \( x = 0 \) and is concave upward for all \( x \).
