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Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$, which is the slope of the secant - line connecting the points $(a,f(a))$ and $(b,f(b))$.

Step2: Calculate average rate of change for $[0,1]$

For the interval $[0,1]$, $a = 0$, $b = 1$, $f(0)=-2$, $f(1)=1$. Then $\frac{f(1)-f(0)}{1 - 0}=\frac{1-(-2)}{1}=\frac{1 + 2}{1}=3$.

Step3: Calculate average rate of change for $[1,4]$

For the interval $[1,4]$, $a = 1$, $b = 4$, $f(1)=1$, $f(4)=-1$. Then $\frac{f(4)-f(1)}{4 - 1}=\frac{-1 - 1}{3}=\frac{-2}{3}$.

Step4: Calculate average rate of change for $[4,5]$

For the interval $[4,5]$, $a = 4$, $b = 5$, $f(4)=-1$, $f(5)=-1$. Then $\frac{f(5)-f(4)}{5 - 4}=\frac{-1-(-1)}{1}=0$.

Step5: Calculate average rate of change for $[5,6]$

For the interval $[5,6]$, $a = 5$, $b = 6$, $f(5)=-1$, $f(6)=-1$. Then $\frac{f(6)-f(5)}{6 - 5}=\frac{-1-(-1)}{1}=0$.

Answer:

The function has the greatest average rate of change over the interval $[0,1]$.