consider the graph of g(x) shown below. (a) if g(x) is the first derivative of f(x), what is the nature of…

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?

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?

Answer

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).