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 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. Aype function \( f \) is concave upward on the subinterval(s) ( \( -\infty \), \( \infty \) ). as needed.) B. The function \( f \) is never concave upward. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) is concave downate answers exact answer. Use a comma 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.
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The Deep Dive
To analyze the function \( f(x) = e^{0.5 x} + 324 e^{-0.5 x} \), we first find its derivative: \( f'(x) = 0.5e^{0.5 x} - 162e^{-0.5 x} \). Setting \( f'(x) \) to zero leads to a critical point, confirming that there is no local minimum. The function is indeed concave upward for all real numbers because the second derivative \( f''(x) = 0.25 e^{0.5 x} + 81 e^{-0.5 x} \) is always positive. In interval notation, the function is concave upward on \( (-\infty, \infty) \) and concave downward on the empty set, meaning it never curves downwards. Therefore, we conclude: there is no local minimum, it's concave upward on \( (-\infty, \infty) \), and it's never concave downward.
