The average rate of change of a function f(x) over some interval [a, b] is the difference quotient,
[tex]\dfrac{f(b)-f(a)}{b-a}[/tex]
which corresponds to the slope of the line connecting the points (a, f(a)) and (b, f(b)) in the graph of f(x).
Given f(x) = 3x² - x³ (correct me if I'm wrong, the exponents look cut off in your screenshot), the average rate of change on [1, 5] is
[tex]\dfrac{f(5)-f(1)}{5-1}=\dfrac{(3\cdot5^2-5^3)-(3\cdot1^2-1^3)}4 = \dfrac{75-125-3+1}4 = \boxed{-13}[/tex]
