part 10 of 10\nsummarize the pertinent information obtained by applying the graphing strategy and sketch the…

Explanation:

Step1: Find the derivative of (f(x))

Use the product - rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 6x) and (v=e^{-0.5x}). (u^\prime=6) and (v^\prime=- 0.5e^{-0.5x}). Then (f^\prime(x)=6e^{-0.5x}-3xe^{-0.5x}=e^{-0.5x}(6 - 3x)).

Step2: Find the critical points

Set (f^\prime(x)=0). Since (e^{-0.5x}\gt0) for all real (x), we solve (6 - 3x = 0), which gives (x = 2).

Step3: Analyze the sign of the derivative

When (x\lt2), (f^\prime(x)=e^{-0.5x}(6 - 3x)\gt0), so (f(x)) is increasing. When (x\gt2), (f^\prime(x)=e^{-0.5x}(6 - 3x)\lt0), so (f(x)) is decreasing. So (x = 2) is a local maximum.

Step4: Find the (y) - value at the local maximum

(f(2)=6\times2\times e^{-0.5\times2}=12e^{-1}\approx4.42).

Step5: Analyze the end - behavior

As (x\to-\infty), (6x\to-\infty) and (e^{-0.5x}\to\infty), so (y = f(x)\to-\infty). As (x\to\infty), the exponential decay (e^{-0.5x}) dominates, and (y = f(x)\to0).

Answer:

D