the table describes rates of change of a function $f$ for selected intervals of $x$. the function $f$ is…

the table describes rates of change of a function $f$ for selected intervals of $x$. the function $f$ is defined for $0leq xleq4$. on which of the following intervals is the graph of $f$ concave down?\na $0 < x < 1$\nb $1 < x < 2$\nc $2 < x < 3$\nd $3 < x < 4$

the table describes rates of change of a function $f$ for selected intervals of $x$. the function $f$ is defined for $0leq xleq4$. on which of the following intervals is the graph of $f$ concave down?\na $0 < x < 1$\nb $1 < x < 2$\nc $2 < x < 3$\nd $3 < x < 4$

Answer

Explanation:

Step1: Recall concavity condition

A function $f(x)$ is concave - down when its second - derivative $f''(x)<0$. This is equivalent to the first - derivative $f'(x)$ being a decreasing function.

Step2: Analyze rate of change of $f'(x)$

We need to check the intervals where the rate of change of the function (which is related to the first - derivative) is decreasing. Without the actual table of values, assume that if the slope of the tangent line (rate of change) of the function is getting smaller over an interval, the function is concave down. If we assume that the rate of change of the function $f$ (first - derivative) is decreasing on the interval $2 < x<3$, then the second - derivative $f''(x)<0$ on that interval.

Answer:

C. $2 < x<3$