sketch a function that is continuous on (-∞,∞) and has the following properties. use a number line to…

sketch a function that is continuous on (-∞,∞) and has the following properties. use a number line to summarize information about the function. f(x)<0 on (-∞,0); f(x)>0 on (0,5); f(x)<0 on (5,∞). which of the following graphs matches the description of the given properties? o a. o b. o c. o d.

sketch a function that is continuous on (-∞,∞) and has the following properties. use a number line to summarize information about the function. f(x)<0 on (-∞,0); f(x)>0 on (0,5); f(x)<0 on (5,∞). which of the following graphs matches the description of the given properties? o a. o b. o c. o d.

Answer

Explanation:

Step1: Analyze the sign of the derivative

If (f^{\prime}(x)<0) on an interval, the function is decreasing on that interval. If (f^{\prime}(x)>0) on an interval, the function is increasing on that interval. Here, (f(x)) is decreasing on ((-\infty,0)), increasing on ((0,5)) and decreasing on ((5,\infty)).

Step2: Check each option

  • Option A: The function is decreasing on ((-\infty,0)), increasing on ((0,5)) and decreasing on ((5,\infty)) which matches the given derivative - sign conditions.
  • Option B: The function is increasing on ((-\infty,0)) which does not match (f^{\prime}(x)<0) on ((-\infty,0)).
  • Option C: The function is decreasing on ((0,5)) which does not match (f^{\prime}(x)>0) on ((0,5)).
  • Option D: The function is increasing on ((5,\infty)) which does not match (f^{\prime}(x)<0) on ((5,\infty)).

Answer:

A.