Question

(d) on what open intervals is f concave up? select the correct choice and, if necessary, fill in the answer 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 - derivative relationship

A function \(y = f(x)\) is concave up when \(f''(x)>0\). Since \(f''(x)\) is the derivative of \(f'(x)\), we need to find where the slope of \(y = f'(x)\) is positive.

Step2: Analyze the graph of \(y = f'(x)\)

Looking at the graph of \(y=f'(x)\), we observe the intervals where the function \(y = f'(x)\) is increasing.
When the graph of \(y = f'(x)\) is increasing, \(f''(x)>0\). From the graph, we can see that \(y = f'(x)\) is increasing on the interval \((2,14)\) (by estimating the \(x -\) values where the slope of \(y = f'(x)\) is positive).

Answer:

A. The graph of \(f\) is concave up on the open interval(s) \((2,14)\)