To determine over which interval the function [tex]\( g(x) = -\frac{x^2}{4} + 7 \)[/tex] has a negative average rate of change, we need to calculate the average rate of change over each interval given in the choices and identify any negative values.
The average rate of change of a function [tex]\( g \)[/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]
Let's compute this for each interval:
### 1. Interval [tex]\([0,4]\)[/tex]
- [tex]\(a = 0\)[/tex], [tex]\(b = 4\)[/tex]
- [tex]\(g(0) = -\frac{0^2}{4} + 7 = 7\)[/tex]
- [tex]\(g(4) = -\frac{4^2}{4} + 7 = -4 + 7 = 3\)[/tex]
The average rate of change is:
[tex]\[
\frac{g(4) - g(0)}{4 - 0} = \frac{3 - 7}{4 - 0} = \frac{-4}{4} = -1
\][/tex]
### 2. Interval [tex]\([-2, 0]\)[/tex]
- [tex]\(a = -2\)[/tex], [tex]\(b = 0\)[/tex]
- [tex]\(g(-2) = -\frac{(-2)^2}{4} + 7 = -1 + 7 = 6\)[/tex]
- [tex]\(g(0) = 7\)[/tex]
The average rate of change is:
[tex]\[
\frac{g(0) - g(-2)}{0 + 2} = \frac{7 - 6}{0 + 2} = \frac{1}{2} = 0.5
\][/tex]
### 3. Interval [tex]\([-8, -4]\)[/tex]
- [tex]\(a = -8\)[/tex], [tex]\(b = -4\)[/tex]
- [tex]\(g(-8) = -\frac{(-8)^2}{4} + 7 = -16 + 7 = -9\)[/tex]
- [tex]\(g(-4) = -\frac{(-4)^2}{4} + 7 = -4 + 7 = 3\)[/tex]
The average rate of change is:
[tex]\[
\frac{g(-4) - g(-8)}{-4 - (-8)} = \frac{3 - (-9)}{-4 + 8} = \frac{3 + 9}{4} = \frac{12}{4} = 3
\][/tex]
### 4. Interval [tex]\([-4, -2]\)[/tex]
- [tex]\(a = -4\)[/tex], [tex]\(b = -2\)[/tex]
- [tex]\(g(-4) = 3\)[/tex]
- [tex]\(g(-2) = 6\)[/tex]
The average rate of change is:
[tex]\[
\frac{g(-2) - g(-4)}{-2 - (-4)} = \frac{6 - 3}{-2 + 4} = \frac{3}{2} = 1.5
\][/tex]
### Identifying the Interval with a Negative Average Rate of Change
From the calculations, the average rates of change for the intervals are:
- [tex]\([0, 4]\)[/tex]: [tex]\(-1\)[/tex]
- [tex]\([-2, 0]\)[/tex]: [tex]\(0.5\)[/tex]
- [tex]\([-8, -4]\)[/tex]: [tex]\(3\)[/tex]
- [tex]\([-4, -2]\)[/tex]: [tex]\(1.5\)[/tex]
The only interval with a negative average rate of change is [tex]\([0, 4]\)[/tex].
Therefore, the correct answer is:
(A) [tex]\([0, 4]\)[/tex]
