sketch a non - constant function that is continuous on (-oo,oo) and has the following properties. use a…

sketch a non - constant function that is continuous on (-oo,oo) and has the following properties. use a number line to summarize information about the function. f(-1)=f(7)=f(-1)=f(3)=f(7)=0; f(x)≥0 on (-oo,oo) which of the following graphs matches the description of the given properties? o a. o b. o c. o d.

Answer

Explanation:

Step1: Analyze given conditions

We know (f(-1)=f(7)), so the function has the same (y -)value at (x = - 1) and (x = 7). Also (f'(-1)=f'(3)=f'(7)=0), which means the function has horizontal - tangents at (x=-1), (x = 3) and (x = 7). And (f(x)\geq0) for all (x\in(-\infty,\infty)), so the function lies on or above the (x -)axis.

Step2: Check option A

In option A, the function dips below the (x -)axis, so it does not satisfy (f(x)\geq0) for all (x\in(-\infty,\infty)).

Step3: Check option B

The function in option B has a negative (y -)value in some intervals, so it does not satisfy (f(x)\geq0) for all (x\in(-\infty,\infty)).

Step4: Check option C

The function in option C has horizontal tangents at approximately (x=-1), (x = 3) and (x = 7), (f(-1)=f(7)) (same (y -)value at these points) and (f(x)\geq0) for all (x\in(-\infty,\infty)).

Step5: Check option D

The function in option D does not have the same (y -)value at (x=-1) and (x = 7).