the graph of f(x) is shown on the interval - 5≤x≤5. determine whether each is less than, greater than or…

Explanation:

Step1: Analyze (f(x)) value

Since the graph of (y = f(x)) lies below the (x -)axis on the interval (-5\leq x\leq5), for all (x) in the interval (-5\leq x\leq5), (f(x)<0).

Step2: Analyze (f^{\prime}(x)) value

The graph of (y = f(x)) is a horizontal - line. The slope of a horizontal line is (0). The first - derivative (f^{\prime}(x)) represents the slope of the tangent line to the graph of (y = f(x)). So (f^{\prime}(x)=0).

Step3: Analyze (f^{\prime\prime}(x)) value

The second - derivative (f^{\prime\prime}(x)) represents the rate of change of the slope (f^{\prime}(x)). Since (f^{\prime}(x)) is a constant function (its value is (0) for all (x) in the interval), the rate of change of (f^{\prime}(x)) is (0), so (f^{\prime\prime}(x)=0).

Answer:

(f(x)<0), (f^{\prime}(x)=0), (f^{\prime\prime}(x)=0)