Given that the function is [tex]f(x)=x^{3}-6 x^{2}+4 x+7[/tex]
We need to determine the average rate of change over the interval [tex]0 \leq x \leq 5[/tex]
Value of f(x) when x = 0:
Substituting x = 0 in the function [tex]f(x)=x^{3}-6 x^{2}+4 x+7[/tex], we have;
[tex]f(0)=(0)^{3}-6 (0)^{2}+4 (0)+7[/tex]
[tex]f(0)=0-0+0+7[/tex]
[tex]f(0)=7[/tex]
Thus, the value of f(0) is 7.
Value of f(x) when x = 5:
Substituting x = 5 in the function [tex]f(x)=x^{3}-6 x^{2}+4 x+7[/tex], we have;
[tex]f(5)=(5)^{3}-6 (5)^{2}+4 (5)+7[/tex]
[tex]f(5)=125-150+20+7[/tex]
[tex]f(5)=2[/tex]
Thus, the value of f(5) is 2.
Average rate of change:
The average rate of change can be determined using the formula,
[tex]Rate \ of \ change=\frac{f(b)-f(a)}{b-a}[/tex]
where [tex]a=0[/tex] and [tex]b=5[/tex]
Thus, we have;
[tex]Rate \ of \ change=\frac{f(5)-f(0)}{5-0}[/tex]
[tex]Rate \ of \ change=\frac{2-7}{5-0}[/tex]
[tex]Rate \ of \ change=\frac{-5}{5}[/tex]
[tex]Rate \ of \ change=-1[/tex]
Thus, the average rate of change over the interval [tex]0 \leq x \leq 5[/tex] is -1.
