To determine over which interval the function [tex]\( g(t) = -(t-1)^2 + 5 \)[/tex] has an average rate of change of zero, we need to follow these steps:
1. Recall the formula for the average rate of change of a function:
The average rate of change of [tex]\( g(t) \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[
\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
\][/tex]
2. Evaluate [tex]\( g(t) \)[/tex] at the interval endpoints for each interval:
Let's look at the intervals one by one and compute [tex]\( g(a) \)[/tex] and [tex]\( g(b) \)[/tex] for [tex]\( a \)[/tex] and [tex]\( b \)[/tex] being the endpoints of the intervals.
### (A) Interval [tex]\([-2, 0]\)[/tex]
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 0 \)[/tex]
[tex]\[
g(-2) = -((-2) - 1)^2 + 5 = -9 + 5 = -4
\][/tex]
[tex]\[
g(0) = -(0 - 1)^2 + 5 = -1 + 5 = 4
\][/tex]
[tex]\[
\text{Average Rate of Change} = \frac{g(0) - g(-2)}{0 - (-2)} = \frac{4 - (-4)}{2} = \frac{4 + 4}{2} = 4
\][/tex]
### (B) Interval [tex]\([-4, -3]\)[/tex]
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = -3 \)[/tex]
[tex]\[
g(-4) = -((-4) - 1)^2 + 5 = -25 + 5 = -20
\][/tex]
[tex]\[
g(-3) = -((-3) - 1)^2 + 5 = -16 + 5 = -11
\][/tex]
[tex]\[
\text{Average Rate of Change} = \frac{g(-3) - g(-4)}{-3 - (-4)} = \frac{-11 - (-20)}{1} = \frac{-11 + 20}{1} = 9
\][/tex]
### (C) Interval [tex]\([-2, 4]\)[/tex]
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 4 \)[/tex]
[tex]\[
g(-2) = -((-2) - 1)^2 + 5 = -9 + 5 = -4
\][/tex]
[tex]\[
g(4) = -(4 - 1)^2 + 5 = -9 + 5 = -4
\][/tex]
[tex]\[
\text{Average Rate of Change} = \frac{g(4) - g(-2)}{4 - (-2)} = \frac{-4 - (-4)}{6} = \frac{-4 + 4}{6} = 0
\][/tex]
### (D) Interval [tex]\([1, 4]\)[/tex]
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
[tex]\[
g(1) = -(1 - 1)^2 + 5 = 5
\][/tex]
[tex]\[
g(4) = -(4 - 1)^2 + 5 = -9 + 5 = -4
\][/tex]
[tex]\[
\text{Average Rate of Change} = \frac{g(4) - g(1)}{4 - 1} = \frac{-4 - 5}{3} = \frac{-9}{3} = -3
\][/tex]
3. Analyze the results:
Among the computed average rates of change, we see that the average rate of change is zero for interval [tex]\([-2, 4]\)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{-2 \leq t \leq 4}
\][/tex]
