Given the graph of f(x), on which interval is the average rate of change the greatest?

1. Understand Average Rate of Change: The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by the formula $\frac{f(b) - f(a)}{b - a}$. This represents the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ on the graph of the function.

2. Identify Points from the Graph: We need to find the coordinates of the points on the graph that correspond to the endpoints of the given intervals. From the graph, we can identify the following points:
- At $x=0$, $f(0) = -2$
- At $x=1$, $f(1) = 1$
- At $x=4$, $f(4) = -1$
- At $x=5$, $f(5) = -1$
- At $x=6$, $f(6) = 0$

3. Calculate Average Rate of Change for Each Interval: Now, we calculate the average rate of change for each of the given intervals:
- Interval [1, 4]: $a=1$, $b=4$. $f(1)=1$, $f(4)=-1$. Average rate of change = $\frac{f(4) - f(1)}{4 - 1} = \frac{-1 - 1}{3} = \frac{-2}{3}$.
- Interval [5, 6]: $a=5$, $b=6$. $f(5)=-1$, $f(6)=0$. Average rate of change = $\frac{f(6) - f(5)}{6 - 5} = \frac{0 - (-1)}{1} = \frac{1}{1} = 1$.
- Interval [0, 1]: $a=0$, $b=1$. $f(0)=-2$, $f(1)=1$. Average rate of change = $\frac{f(1) - f(0)}{1 - 0} = \frac{1 - (-2)}{1} = \frac{3}{1} = 3$.
- Interval [4, 5]: $a=4$, $b=5$. $f(4)=-1$, $f(5)=-1$. Average rate of change = $\frac{f(5) - f(4)}{5 - 4} = \frac{-1 - (-1)}{1} = \frac{0}{1} = 0$.

4. Compare the Average Rates of Change: We compare the calculated average rates of change:
- Interval [1, 4]: $-\frac{2}{3}$
- Interval [5, 6]: $1$
- Interval [0, 1]: $3$
- Interval [4, 5]: $0$

1. Determine the Greatest Average Rate of Change: The greatest value among these is $3$, which corresponds to the interval $[0, 1]$.