Question

on what open intervals is f concave up? select the correct choice and, if necessary, fill in the swer box to complete your choice.
a. the graph of f is concave up on the open interval(s) (type your answer in interval notation. round to the nearest integer as needed. use a comma to separate answers as needed. )
b. the graph of f is not concave up on any open interval.

Explanation:

Step1: Recall the concavity rule

If \(y = f(x)\) is a differentiable function, then \(f(x)\) is concave up when \(f^{\prime\prime}(x)>0\). Also, by the relationship between the first - derivative and the second - derivative, if \(f^{\prime}(x)\) is increasing, then \(f^{\prime\prime}(x)>0\).

Step2: Analyze the graph of \(y = f^{\prime}(x)\)

We look for the intervals where the graph of \(y = f^{\prime}(x)\) is increasing.
From the graph of \(y = f^{\prime}(x)\), we can observe that the function \(y = f^{\prime}(x)\) is increasing on the intervals \((-20,-15)\) and \((0,5)\) (rounded to the nearest integer).

Answer:

A. The graph of \(f\) is concave up on the open interval(s) \((-20,-15),(0,5)\)