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 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 downward on the subinterval(s) (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as. 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 $$ x= $$ (Type an exact answer. Use a comma to separate answers as needed.) B. The function $$ f $$ has no inflection point.

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1. **Function and its derivatives**

The function is 
   $$
   f(x)=e^{0.5x}+324\,e^{-0.5x}.
   $$
   
   The first derivative is obtained by differentiating term‐by‐term:
   $$
   f'(x)=0.5\,e^{0.5x}-162\,e^{-0.5x}.
   $$
   
   The second derivative is
   $$
   f''(x)=0.25\,e^{0.5x}+81\,e^{-0.5x}.
   $$

2. **Concavity analysis**

Since the exponential function is positive for all $$x$$, both terms in
   $$
   f''(x)=0.25\,e^{0.5x}+81\,e^{-0.5x}
   $$
   are positive. Hence,
   $$
   f''(x)>0\quad\text{for all }x.
   $$
   This implies that the function is always concave upward and is never concave downward.
   
   Therefore, from the options provided:
   
   - For concavity:  
     **B. The function $$ f $$ is never concave downward.**

3. **Inflection point analysis**

An inflection point occurs when the concavity of the function changes, i.e., when $$ f''(x) $$ changes sign. Since
   $$
   f''(x)>0\quad\text{for all }x,
   $$
   there is no change in concavity and thus no inflection point.
   
   Therefore, from the options provided:
   
   - For inflection points:  
     **B. The function $$ f $$ has no inflection point.**

4. **Additional information from graphing**

- **Behavior at infinity:**  
     As $$ x\to\infty $$, the term $$ e^{0.5x} $$ dominates, so $$ f(x) \to \infty $$.  
     As $$ x\to -\infty $$, the term $$ 324\,e^{-0.5x} $$ dominates, so $$ f(x) \to \infty $$.

- **Finding the minimum:**  
     To locate the minimum, set the first derivative equal to zero:
     $$
     0.5\,e^{0.5x}-162\,e^{-0.5x}=0.
     $$
     Solving for $$ x $$:
     $$
     \begin{aligned}
     0.5\,e^{0.5x} &= 162\,e^{-0.5x} \
     e^{0.5x}\cdot e^{0.5x} &= 324 \quad (\text{multiplying both sides by }e^{0.5x}) \
     e^x &= 324 \
     x &= \ln(324).
     \end{aligned}
     $$
     Substitute $$ x=\ln(324) $$ into $$ f(x) $$:
     $$
     \begin{aligned}
     f(\ln(324)) &= e^{0.5\ln(324)}+324\,e^{-0.5\ln(324)} \
     &= 324^{0.5}+324\,(324^{-0.5}) \
     &= 18+18=36.
     \end{aligned}
     $$
     Hence, there is a minimum at the point $$ \bigl(\ln(324),\,36\bigr) $$.

5. **Summary of Answers**

- Concavity:  
     **B. The function $$ f $$ is never concave downward.**

- Inflection points:  
     **B. The function $$ f $$ has no inflection point.**

6. **Sketching the graph**

The graph of $$ y=f(x) $$ has the following characteristics:
   
   - It has a minimum point at $$ \left(\ln(324),\,36\right) $$.
   - As $$ x\to-\infty $$ and $$ x\to\infty $$, the function tends to infinity.
   - The function is always concave upward (convex) since $$ f''(x)>0 $$ for all $$ x $$.

The sketch should include a smooth curve that descends to the minimum point at $$ \left(\ln(324),\,36\right) $$ and then increases on both sides, remaining concave upward everywhere.
The function $$ f(x) = e^{0.5x} + 324e^{-0.5x} $$ is always concave upward and has no inflection points. It has a minimum at $$ x = \ln(324) $$ with a value of 36.

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