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To determine over which interval the function [tex]\( f(x) = x^2 - x - 1 \)[/tex] has an average rate of change of zero, let's analyze each given interval and calculate the average rate of change step-by-step.
The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
Let's analyze the intervals provided.
### Interval A: [tex]\(-5 \leq x \leq 5\)[/tex]
[tex]\[ a = -5, \quad b = 5 \][/tex]
[tex]\[ f(a) = f(-5) = (-5)^2 - (-5) - 1 = 25 + 5 - 1 = 29 \][/tex]
[tex]\[ f(b) = f(5) = 5^2 - 5 - 1 = 25 - 5 - 1 = 19 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(5) - f(-5)}{5 - (-5)} = \frac{19 - 29}{10} = \frac{-10}{10} = -1 \][/tex]
The average rate of change over interval A is [tex]\(-1\)[/tex], which is not zero.
### Interval B: [tex]\(-3 \leq x \leq -2\)[/tex]
[tex]\[ a = -3, \quad b = -2 \][/tex]
[tex]\[ f(a) = f(-3) = (-3)^2 - (-3) - 1 = 9 + 3 - 1 = 11 \][/tex]
[tex]\[ f(b) = f(-2) = (-2)^2 - (-2) - 1 = 4 + 2 - 1 = 5 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{5 - 11}{1} = -6 \][/tex]
The average rate of change over interval B is [tex]\(-6\)[/tex], which is not zero.
### Interval C: [tex]\(-1 \leq x \leq 2\)[/tex]
[tex]\[ a = -1, \quad b = 2 \][/tex]
[tex]\[ f(a) = f(-1) = (-1)^2 - (-1) - 1 = 1 + 1 - 1 = 1 \][/tex]
[tex]\[ f(b) = f(2) = 2^2 - 2 - 1 = 4 - 2 - 1 = 1 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{1 - 1}{3} = \frac{0}{3} = 0 \][/tex]
The average rate of change over interval C is [tex]\(0\)[/tex], which is exactly what we are looking for.
### Interval D: [tex]\(2 \leq x \leq 3\)[/tex]
[tex]\[ a = 2, \quad b = 3 \][/tex]
[tex]\[ f(a) = f(2) = 2^2 - 2 - 1 = 1 \][/tex]
[tex]\[ f(b) = f(3) = 3^2 - 3 - 1 = 9 - 3 - 1 = 5 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(3) - f(2)}{3 - 2} = \frac{5 - 1}{1} = 4 \][/tex]
The average rate of change over interval D is [tex]\(4\)[/tex], which is not zero.
After analyzing all the intervals, we conclude:
The function [tex]\( f(x) = x^2 - x - 1 \)[/tex] has an average rate of change of zero over the interval:
[tex]\[ \boxed{-1 \leq x \leq 2} \][/tex]
So the correct answer is (C).
The formula for the average rate of change of a function [tex]\( f(x) \)[/tex] over the interval [tex]\([a, b]\)[/tex] is given by:
[tex]\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \][/tex]
Let's analyze the intervals provided.
### Interval A: [tex]\(-5 \leq x \leq 5\)[/tex]
[tex]\[ a = -5, \quad b = 5 \][/tex]
[tex]\[ f(a) = f(-5) = (-5)^2 - (-5) - 1 = 25 + 5 - 1 = 29 \][/tex]
[tex]\[ f(b) = f(5) = 5^2 - 5 - 1 = 25 - 5 - 1 = 19 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(5) - f(-5)}{5 - (-5)} = \frac{19 - 29}{10} = \frac{-10}{10} = -1 \][/tex]
The average rate of change over interval A is [tex]\(-1\)[/tex], which is not zero.
### Interval B: [tex]\(-3 \leq x \leq -2\)[/tex]
[tex]\[ a = -3, \quad b = -2 \][/tex]
[tex]\[ f(a) = f(-3) = (-3)^2 - (-3) - 1 = 9 + 3 - 1 = 11 \][/tex]
[tex]\[ f(b) = f(-2) = (-2)^2 - (-2) - 1 = 4 + 2 - 1 = 5 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(-2) - f(-3)}{-2 - (-3)} = \frac{5 - 11}{1} = -6 \][/tex]
The average rate of change over interval B is [tex]\(-6\)[/tex], which is not zero.
### Interval C: [tex]\(-1 \leq x \leq 2\)[/tex]
[tex]\[ a = -1, \quad b = 2 \][/tex]
[tex]\[ f(a) = f(-1) = (-1)^2 - (-1) - 1 = 1 + 1 - 1 = 1 \][/tex]
[tex]\[ f(b) = f(2) = 2^2 - 2 - 1 = 4 - 2 - 1 = 1 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(2) - f(-1)}{2 - (-1)} = \frac{1 - 1}{3} = \frac{0}{3} = 0 \][/tex]
The average rate of change over interval C is [tex]\(0\)[/tex], which is exactly what we are looking for.
### Interval D: [tex]\(2 \leq x \leq 3\)[/tex]
[tex]\[ a = 2, \quad b = 3 \][/tex]
[tex]\[ f(a) = f(2) = 2^2 - 2 - 1 = 1 \][/tex]
[tex]\[ f(b) = f(3) = 3^2 - 3 - 1 = 9 - 3 - 1 = 5 \][/tex]
[tex]\[ \text{Average rate of change} = \frac{f(3) - f(2)}{3 - 2} = \frac{5 - 1}{1} = 4 \][/tex]
The average rate of change over interval D is [tex]\(4\)[/tex], which is not zero.
After analyzing all the intervals, we conclude:
The function [tex]\( f(x) = x^2 - x - 1 \)[/tex] has an average rate of change of zero over the interval:
[tex]\[ \boxed{-1 \leq x \leq 2} \][/tex]
So the correct answer is (C).
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