Question

the following is a graph of the first derivative of the function f(x). on which of the following intervals is f(x) concave down?

Explanation:

Step1: Recall concavity - derivative relationship

A function $f(x)$ is concave down when $f''(x)<0$. Since the given graph is of $f'(x)$, we need to find where $f'(x)$ is decreasing (because the derivative of $f'(x)$ is $f''(x)$).

Step2: Analyze the graph

Looking at the graph of $f'(x)$, we see that it is decreasing on the intervals where the slope of $f'(x)$ is negative. This occurs on the intervals from $x = - 1$ to $x=0$ and from $x = 0$ to $x = 1$. In terms of the labeled points, this is the interval from $D$ to $G$.

Answer:

The function $f(x)$ is concave down on the interval from the $x$-coordinate of point $D$ to the $x$-coordinate of point $G$.