Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=e^{0.5 x}+324 e^{-0.5 x} \) B. The function \( f \) is never increasing. (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers A. The function \( f \) is decreasing on the subinterval(s) ( \( -\infty \), In ( 324\( ) \) ). (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The function \( f \) is never decreasing. lect the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function \( f \) has a local maximum at \( x=\square \) if necessary, fill in the answer box to complete your choice. (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function \( f \) has no local maximum.
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To explore the function \( f(x) = e^{0.5x} + 324e^{-0.5x} \), we start by analyzing its behavior. First, we can find the critical points by calculating the derivative \( f'(x) \), then setting it to zero. This leads us to the conclusion that \( f \) is decreasing on the interval \( (-\infty, \ln(324)) \) and is never increasing elsewhere. When it comes to local extrema, the absence of any critical points where the derivative equals zero confirms that this function doesn’t possess a local maximum. Therefore, we state that the function \( f \) has no local maximum. In short, the useful intervals here are: **Decreasing Interval**: \( (-\infty, \ln(324)) \) **Local Maximum**: None.
