Question

the graph of f(x) is shown on the interval - 5≤x≤5. determine whether each is less than, greater than or equal to 0. assume f is twice differentiable. f(x) ▼ 0 f(x) ▼ 0 f(x) ▼ 0

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:

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