Answer :
To find the average rate of change of the function
[tex]$$
f(x) = x^4 - 5x
$$[/tex]
on the interval [tex]$[0, 3]$[/tex], we use the formula for the average rate of change:
[tex]$$
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a},
$$[/tex]
where [tex]$a$[/tex] and [tex]$b$[/tex] are the endpoints of the interval, in this case [tex]$a = 0$[/tex] and [tex]$b = 3$[/tex].
Step 1: Calculate [tex]$f(0)$[/tex].
[tex]$$
f(0) = 0^4 - 5(0) = 0.
$$[/tex]
Step 2: Calculate [tex]$f(3)$[/tex].
[tex]$$
f(3) = 3^4 - 5(3) = 81 - 15 = 66.
$$[/tex]
Step 3: Substitute [tex]$f(0)$[/tex], [tex]$f(3)$[/tex], and the endpoints into the formula.
[tex]$$
\text{Average Rate of Change} = \frac{f(3) - f(0)}{3 - 0} = \frac{66 - 0}{3} = \frac{66}{3} = 22.
$$[/tex]
Thus, the average rate of change of the function on the interval [tex]$[0, 3]$[/tex] is [tex]$22$[/tex], which corresponds to option (C).
[tex]$$
f(x) = x^4 - 5x
$$[/tex]
on the interval [tex]$[0, 3]$[/tex], we use the formula for the average rate of change:
[tex]$$
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a},
$$[/tex]
where [tex]$a$[/tex] and [tex]$b$[/tex] are the endpoints of the interval, in this case [tex]$a = 0$[/tex] and [tex]$b = 3$[/tex].
Step 1: Calculate [tex]$f(0)$[/tex].
[tex]$$
f(0) = 0^4 - 5(0) = 0.
$$[/tex]
Step 2: Calculate [tex]$f(3)$[/tex].
[tex]$$
f(3) = 3^4 - 5(3) = 81 - 15 = 66.
$$[/tex]
Step 3: Substitute [tex]$f(0)$[/tex], [tex]$f(3)$[/tex], and the endpoints into the formula.
[tex]$$
\text{Average Rate of Change} = \frac{f(3) - f(0)}{3 - 0} = \frac{66 - 0}{3} = \frac{66}{3} = 22.
$$[/tex]
Thus, the average rate of change of the function on the interval [tex]$[0, 3]$[/tex] is [tex]$22$[/tex], which corresponds to option (C).
