Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ y=f(x) $$. $$ f(x)=e^{0.5 x}+324 e^{-0.5 x} $$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ x $$-intercept(s) of $$ f $$ is $$ x= $$ $$ \square $$ (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.) B. The function $$ f $$ has no $$ x $$-intercepts. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The $$ y $$-intercept of $$ f $$ is $$ y=325 $$. (Type an exact answer, using radicals as needed.) B. The function $$ f $$ has no $$ y $$-intercept. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The function $$ f $$ is increasing on the subinterval(s) $$ \square $$ (Type your answer in interval notation. Type an exact answer. Use a comma to separate answers as needed.) B. The function $$ f $$ is never increasing.

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**1. $$ x $$-Intercepts**

We set  
$$
f(x)=e^{0.5x}+324\,e^{-0.5x}=0.
$$
Since $$ e^{0.5x}>0 $$ and $$ 324\,e^{-0.5x}>0 $$ for all real $$ x $$, their sum is always positive. Thus, the equation has no solution.

**Answer:** B. The function $$ f $$ has no $$ x $$-intercepts.

---

**2. $$ y $$-Intercept**

The $$ y $$-intercept is found by evaluating $$ f(0) $$:
$$
f(0)=e^{0}+324\,e^{0}=1+324=325.
$$

**Answer:** A. The $$ y $$-intercept of $$ f $$ is $$ y=325 $$.

---

**3. Increasing Intervals**

First, differentiate $$ f(x) $$:
$$
f'(x)=0.5\,e^{0.5x}-0.5\,(324)\,e^{-0.5x}=0.5\left(e^{0.5x}-324\,e^{-0.5x}\right).
$$
Set $$ f'(x)=0 $$ to find critical points:
$$
0.5\left(e^{0.5x}-324\,e^{-0.5x}\right)=0 \quad\Longrightarrow\quad e^{0.5x}=324\,e^{-0.5x}.
$$
Multiplying both sides by $$ e^{0.5x} $$ (which is positive),
$$
e^{x}=324 \quad\Longrightarrow\quad x=\ln(324).
$$

To determine where $$ f $$ is increasing, analyze the sign of $$ f'(x) $$. Notice:
$$
f'(x)>0 \quad\Longleftrightarrow\quad e^{0.5x}-324\,e^{-0.5x}>0.
$$
Multiplying by $$ e^{0.5x}>0 $$,
$$
e^{x}-324>0 \quad\Longleftrightarrow\quad x>\ln(324).
$$
Thus, the function is increasing on the interval:
$$
(\ln(324),\infty).
$$

**Answer:** A. The function $$ f $$ is increasing on the subinterval(s) $$ (\ln(324),\infty) $$.

---

**Graphing Summary of $$ y=f(x) $$**

- **Domain:** All real numbers.
- **$$ x $$-Intercepts:** None.
- **$$ y $$-Intercept:** $$ (0,325) $$.
- **Critical Point:** The function has a critical point at $$ x=\ln(324) $$ where it reaches a minimum.
- **Monotonicity:** The function is decreasing on $$ (-\infty, \ln(324)) $$ and increasing on $$ (\ln(324),\infty) $$.
- **End Behavior:** As $$ x \to \infty $$, $$ e^{0.5x} $$ dominates and $$ f(x) \to \infty $$; as $$ x \to -\infty $$, $$ e^{-0.5x} $$ dominates and $$ f(x) \to \infty $$ as well.
- **Concavity:** The second derivative is positive for all $$ x $$, so the graph is concave up.

When sketching the graph:
- Plot the $$ y $$-intercept at $$ (0,325) $$.
- Mark the minimum point at $$ \left(\ln(324),\,\text{minimum value}\right) $$ (the exact minimum value can be computed by plugging $$ x=\ln(324) $$ into $$ f(x) $$).
- Show that the graph has no $$ x $$-intercepts.
- Draw the curve decreasing to a minimum and then increasing, with both ends tending to infinity, and note its concave upward nature.

This completes the solution.
**Summary of $$ y=f(x) $$:** - **$$ y $$-Intercept:** $$ (0, 325) $$ - **$$ x $$-Intercepts:** None - **Increasing Interval:** $$ (\ln(324), \infty) $$ - **Graph:** Starts at $$ (0, 325) $$, decreases to a minimum at $$ x = \ln(324) $$, then increases, with both ends approaching infinity. No $$ x $$-intercepts and concave upward throughout.

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