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 \( x \).

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 \( x \).