Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=5 x e^{-0.5 x} $$ 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. **Function Analysis and Domain**  
   The function is  
   $$
   f(x)=5xe^{-0.5x}.
   $$  
   Since the exponential function $$e^{-0.5x}$$ is defined for all real numbers and the factor $$5x$$ is also defined for all real $$x$$, the domain of $$f(x)$$ is all real numbers.  
   **Answer:** B. The domain is all real $$ x $$.

2. **Intercepts**  
   - **x-intercept:** Set $$ f(x)=0 $$.  
     $$
     5xe^{-0.5x}=0 \quad \Longrightarrow \quad x=0.
     $$  
     Thus, the graph intersects the $$x$$-axis at $$(0,0)$$.  
   - **y-intercept:** Evaluate $$ f(0) $$.  
     $$
     f(0)=5\cdot 0\cdot e^{-0.5\cdot 0}=0.
     $$  
     The graph also intersects the $$y$$-axis at $$(0,0)$$.

3. **Critical Points and Extrema**  
   To find the critical points, compute the derivative $$ f'(x) $$.  
   Using the product rule:  
   $$
   f'(x)=\frac{d}{dx}[5x]\cdot e^{-0.5x}+5x\cdot\frac{d}{dx}[e^{-0.5x}].
   $$  
   We have  
   $$
   \frac{d}{dx}[5x]=5 \quad \text{and} \quad \frac{d}{dx}[e^{-0.5x}]=-0.5e^{-0.5x}.
   $$  
   Thus,  
   $$
   f'(x)=5e^{-0.5x}+5x\cdot(-0.5e^{-0.5x})=5e^{-0.5x}\left(1-0.5x\right).
   $$  
   Set $$ f'(x)=0 $$:  
   $$
   5e^{-0.5x}\left(1-0.5x\right)=0.
   $$  
   Since $$5e^{-0.5x}$$ is never zero, we require  
   $$
   1-0.5x=0 \quad \Longrightarrow \quad x=2.
   $$  
   At $$ x=2 $$,  
   $$
   f(2)=5\cdot2\cdot e^{-1}=\frac{10}{e}\approx3.678.
   $$  
   Therefore, there is a local (and global) maximum at $$(2, 10/e)$$.

4. **Behavior at Infinity**  
   - As $$ x\to\infty $$: The factor $$ e^{-0.5x} $$ decays to 0 faster than $$ x $$ increases, so  
     $$
     f(x)\to 0.
     $$  
   - As $$ x\to-\infty $$: The exponential $$ e^{-0.5x} $$ grows unbounded while $$ x $$ is large negative, hence  
     $$
     f(x)\to -\infty.
     $$

5. **Graph Sketch Summary**  
   - The graph passes through the origin $$(0,0)$$.  
   - It has a global maximum at $$(2, \frac{10}{e})$$.  
   - For $$ x>2 $$, the function decreases toward 0.  
   - For $$ x<0 $$, the function decreases rapidly to $$-\infty$$ as $$ x\to-\infty $$.

The domain of the function is all real $$ x $$.
The domain of $$ f(x) = 5x e^{-0.5x} $$ is all real numbers.

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