the graph of y = f(x) is shown. remember this is the graph of y = f(x), not the graph of y = f(x) based on…

Explanation:

Step1: Recall the first - derivative test

The first - derivative test states that if (y = f'(x)) changes sign from positive to negative at a point (x = c), then (y = f(x)) has a relative maximum at (x = c). If (y = f'(x)) changes sign from negative to positive at a point (x = c), then (y = f(x)) has a relative minimum at (x = c).

Step2: Analyze the sign - change of (y = f'(x)) for relative maximum

Looking at the graph of (y = f'(x)), we see that (f'(x)) changes sign from positive to negative at (x=-1). Before (x = - 1), (f'(x)>0) (the graph of (y = f'(x)) is above the (x) - axis) and after (x=-1), (f'(x)<0) (the graph of (y = f'(x)) is below the (x) - axis). So (y = f(x)) has a relative maximum at (x=-1).

Step3: Analyze the sign - change of (y = f'(x)) for relative minimum

We observe that (f'(x)) changes sign from negative to positive at (x = 1). Before (x = 1), (f'(x)<0) (the graph of (y = f'(x)) is below the (x) - axis) and after (x = 1), (f'(x)>0) (the graph of (y = f'(x)) is above the (x) - axis). So (y = f(x)) has a relative minimum at (x = 1).

Answer:

(y = f(x)) has a relative maximum at (x=-1) (y = f(x)) has a relative minimum at (x = 1)