Question

the table describes rates of change of a function f for selected intervals of x. the function f is defined for 0 ≤ x ≤ 4. on which of the following intervals is the graph of f concave down? a 0 < x < 1 b 1 < x < 2 c 2 < x < 3 d 3 < x < 4

Explanation:

Step1: Recall concavity definition

A function $f(x)$ is concave - down when its second - derivative $f''(x)<0$. This means that the first - derivative $f'(x)$ is decreasing. In terms of rates of change, the rate of change of the function is decreasing.

Step2: Analyze intervals

We need to look at the intervals and check where the rate of change of the function is decreasing. Without the actual table of rates of change (but based on the concept), we know that when the slope of the tangent line to the function (which is the rate of change) is getting smaller as $x$ increases, the function is concave down.

Answer:

We need the table of rates of change to determine the correct interval. Since the table is not provided, we cannot give a definite answer from the options A. $0 < x<1$, B. $1 < x<2$, C. $2 < x<3$, D. $3 < x<4$.