The average rate of change of f(x) between 1 and 4 is computed as follows:
[tex]\frac{f(4)-f(1)}{4-1}[/tex]f(x) evaluated at x = 4 and x = 1 are computed as follows:
[tex]\begin{gathered} f(x)=\frac{1}{2}x^2-3 \\ f(4)=\frac{1}{2}4^2-3=5 \\ f(1)=\frac{1}{2}1^2-3=-\frac{5}{2} \end{gathered}[/tex]Then, the average rate of change is:
[tex]\frac{5-(-\frac{5}{2})}{4-1}=\frac{\frac{15}{2}}{3}=\frac{15}{6}[/tex]