Given:
[tex]\begin{gathered} f(x)=8x^2+x+3 \\ g(x)=4^x-1 \\ h(x)=3x+6 \end{gathered}[/tex]
a) the average rate of change in the interval [3,5] is,
[tex]\begin{gathered} f_{avg}=\frac{f(5)-f(3)}{5-3} \\ f_{avg}=\frac{8(5^2)+5+3-(8(3^2)+3+3)}{2} \\ f_{avg}=\frac{130}{2}=65 \end{gathered}[/tex]
Also,
[tex]\begin{gathered} g_{avg}=\frac{g(5)-g(3)}{5-3} \\ g_{avg}=\frac{4^5-1-(4^3-1)}{2} \\ g_{avg}=480 \end{gathered}[/tex]
And,
[tex]\begin{gathered} h_{avg}=\frac{h(5)-h(3)}{5-3} \\ h_{avg}=\frac{3(5)+6-(3(3)+6))}{2} \\ h_{avg}=\frac{6}{2}=3 \end{gathered}[/tex]
Option a) is not correct. because the average rate of change of g and h is not more than f.
b) for the interval [0,2],
[tex]\begin{gathered} f_{avg}=\frac{f(2)-f(0)}{2-0} \\ =\frac{8(2^2)+2+3-(3)}{2} \\ =17 \\ g_{avg}=\frac{g(2)-g(0)}{2-0} \\ =\frac{4^2-1-(0-1)}{2} \\ =8 \\ h_{avg}=\frac{h(2)-h(0)}{2-0} \\ =\frac{3(2)+6-(6)}{2} \\ =3 \end{gathered}[/tex]
Option b) is also not correct.
when x approaches to infinity, the values of g(x) and h(x) exceeds the values of f(x).
Answer: option d)
