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 number line summarizes the information about the function?

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 number line summarizes the information about the function?

Answer

Explanation:

Step1: Recall derivative - function relationship

If (f^{\prime}(x)<0), the function (f(x)) is decreasing. If (f^{\prime}(x)>0), the function (f(x)) is increasing.

Step2: Analyze the given intervals

We are given (f^{\prime}(x)<0) on ((-\infty,0)), so (f(x)) is decreasing on ((-\infty,0)). (f^{\prime}(x)>0) on ((0,5)), so (f(x)) is increasing on ((0,5)). And (f^{\prime}(x)<0) on ((5,\infty)), so (f(x)) is decreasing on ((5,\infty)).

Step3: Analyze the graphs

For the graph of the function: A decreasing - increasing - decreasing pattern is expected.

Step4: Analyze the number - line

The number - line should have (f^{\prime}(x)<0) on ((-\infty,0)), (f^{\prime}(x)>0) on ((0,5)) and (f^{\prime}(x)<0) on ((5,\infty)).

Answer:

The correct graph is A (since it shows a decreasing function on ((-\infty,0)), an increasing function on ((0,5)) and a decreasing function on ((5,\infty))) and the correct number - line is B (since it has (f^{\prime}(x)<0) on ((-\infty,0)), (f^{\prime}(x)>0) on ((0,5)) and (f^{\prime}(x)<0) on ((5,\infty)))