Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=7 x e^{-0.5 x} $$ Find the domain of $$ f(x) $$. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The domain is all real $$ x $$, except $$ x= $$ (Type an integer or a decimal. Use a comma to separate answers as needed.) B. The domain is all real $$ x $$

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1. **Determining the Domain**

The function is 
   $$
   f(x)=7x\,e^{-0.5x}.
   $$
   Since the exponential function $$ e^{-0.5x} $$ is defined for all real $$ x $$ and the factor $$ 7x $$ is also defined for all real $$ x $$, there is no restriction on the values of $$ x $$.

**Domain:** All real $$ x $$.

2. **Summarizing Information from a Graphing Strategy**

- **Intercepts:**
     - $$ f(0)=7\cdot0\cdot e^{-0.5\cdot0} = 0 $$, so the function passes through the origin.
   
   - **Critical Point and Maximum:**
     - Differentiate $$ f(x) $$ using the product rule:
       $$
       f'(x)=7e^{-0.5x}\,(1-0.5x).
       $$
     - Setting $$ f'(x)=0 $$ gives
       $$
       1-0.5x=0 \quad \Longrightarrow \quad x=2.
       $$
     - At $$ x=2 $$, the function attains a local maximum:
       $$
       f(2)=7\cdot2\cdot e^{-1}=\frac{14}{e} \approx 5.15.
       $$
   
   - **End Behavior:**
     - As $$ x\to \infty $$, the exponential decay $$ e^{-0.5x} $$ dominates, so $$ f(x)\to 0 $$ from above.
     - As $$ x \to -\infty $$, note that $$ e^{-0.5x}= e^{0.5|x|} $$ grows exponentially and since $$ x $$ is negative, $$ f(x) $$ tends to $$-\infty $$.
   
   - **General Shape:**
     - For $$ x>0 $$, the function increases from $$ 0 $$ to its maximum at $$ x=2 $$, then decays toward $$ 0 $$ as $$ x $$ increases.
     - For $$ x<0 $$, the function decreases without bound.

3. **Sketching the Graph**

- Plot the intercept at the origin $$(0,0)$$.
   - Mark the local maximum at $$(2, 14/e)$$.
   - Draw the curve that increases from the origin to the maximum, then decreases toward $$ 0 $$ as $$ x $$ increases.
   - For $$ x<0 $$, draw the curve decreasing steeply toward $$-\infty$$.

4. **Answer to the Domain Question**

The correct choice is:

**B. The domain is all real $$ x $$**.
The domain of $$ f(x) = 7x e^{-0.5x} $$ is all real numbers.

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