1. To determine the average rate of change of a function "f(x)" between the points "x = a" and "x = b" we use the following formula:
[tex]A=\frac{f(b)-f(a)}{b-a}[/tex]2. In this case, we have the following function:
[tex]f(x)=5^x+1[/tex]And we have the points:
[tex]\begin{gathered} x=0 \\ x=4 \end{gathered}[/tex]Now we determine the value of f(b) by replacing x = 4 in the function:
[tex]\begin{gathered} f(4)=5^4+1 \\ f(4)=626 \end{gathered}[/tex]Now we determine f(0):
[tex]\begin{gathered} f(0)=5^0+1 \\ f(0)=1+1=2 \end{gathered}[/tex]Replacing in the formula for the average rate of change we get:
[tex]A=\frac{626-2}{4-0}[/tex]Solving the operations:
[tex]A=\frac{624}{4}=156[/tex]Therefore, the average rate of change is 156.