summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of…
summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of f(x)=6xe^(-0.5x). find the intervals where f(x) is increasing or decreasing. select the correct choice below and fill in the answer box(es) to complete your choice. (type your answer in interval notation. use integers or decimals for any numbers in the expression. use a comma to separate answers as needed.) a. the function is increasing on. it is never decreasing. b. the function is increasing on. it is decreasing on. c. the function is decreasing on. it is never increasing.
Answer
Explanation:
Step1: Find the derivative of (y = f(x)=6xe^{- 0.5x})
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 (y^\prime=6e^{-0.5x}+6x(-0.5e^{-0.5x})=6e^{-0.5x}(1 - 0.5x)).
Step2: Find the critical points
Set (y^\prime = 0). Since (e^{-0.5x}\gt0) for all real (x), we solve (1 - 0.5x = 0). [ \begin{align*} 1-0.5x&=0\ 0.5x&=1\ x& = 2 \end{align*} ]
Step3: Test the intervals
Choose a test - point in the interval ((-\infty,2)), say (x = 0). Then (y^\prime(0)=6e^{0}(1-0)=6\gt0), so the function is increasing on ((-\infty,2)). Choose a test - point in the interval ((2,\infty)), say (x = 3). Then (y^\prime(3)=6e^{-1.5}(1 - 1.5)=6e^{-1.5}(-0.5)\lt0), so the function is decreasing on ((2,\infty)).
Answer:
B. The function is increasing on ((-\infty,2)). It is decreasing on ((2,\infty))