Question

consider the graph of g(x) shown below. (a) if g(x) is the first derivative of f(x), what is the nature of f(x) when x = 3? (b) if g(x) is the second derivative of f(x), what is the nature of f(x) when x = 1?

Explanation:

Step1: Recall derivative - function relationship for part (a)

If \(g(x)=f^{\prime}(x)\), when \(x = 3\), observe the sign of \(g(3)\). From the graph, \(g(3)<0\). Since \(f^{\prime}(x)=g(x)\), when \(f^{\prime}(x)<0\), the function \(f(x)\) is decreasing.

Step2: Recall second - derivative function relationship for part (b)

If \(g(x)=f^{\prime\prime}(x)\), when \(x = 1\), observe the sign of \(g(1)\). From the graph, \(g(1)>0\). Since \(f^{\prime\prime}(x)=g(x)\), when \(f^{\prime\prime}(x)>0\), the function \(f(x)\) is concave - up.

Answer:

(a) \(f(x)\) is decreasing at \(x = 3\).
(b) \(f(x)\) is concave - up at \(x = 1\).