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Sagot :
To determine which statement is true about the functions [tex]\( f(x) = x^3 + 5x^2 - x \)[/tex] and [tex]\( g(x) \)[/tex] over the interval [tex]\([-2, 2]\)[/tex], we need to analyze their behavior and compute the average rates of change.
### Step 1: Evaluate [tex]\( f(x) \)[/tex] at each given [tex]\( x \)[/tex] value.
Given the values of [tex]\( x \)[/tex] are [tex]\([-2, -1, 0, 1, 2]\)[/tex], we compute [tex]\( f(x) \)[/tex]:
- [tex]\( f(-2) = (-2)^3 + 5(-2)^2 - (-2) = -8 + 20 + 2 = 14 \)[/tex]
- [tex]\( f(-1) = (-1)^3 + 5(-1)^2 - (-1) = -1 + 5 + 1 = 5 \)[/tex]
- [tex]\( f(0) = 0^3 + 5(0)^2 - 0 = 0 \)[/tex]
- [tex]\( f(1) = 1^3 + 5(1)^2 - 1 = 1 + 5 - 1 = 5 \)[/tex]
- [tex]\( f(2) = 2^3 + 5(2)^2 - 2 = 8 + 20 - 2 = 26 \)[/tex]
So, the values of [tex]\( f(x) \)[/tex] at [tex]\([-2, -1, 0, 1, 2]\)[/tex] are [tex]\( [14, 5, 0, 5, 26] \)[/tex].
### Step 2: Use the provided values of [tex]\( g(x) \)[/tex]
From the given table, we have the values of [tex]\( g(x) \)[/tex] at [tex]\([-2, -1, 0, 1, 2]\)[/tex] as [tex]\([-4, 8, 6, 2, -16]\)[/tex].
### Step 3: Calculate the rate of change for each function over the interval [tex]\([-2, 2]\)[/tex]
- For [tex]\(\Delta f\)[/tex] (differences in [tex]\( f(x) \)[/tex]):
- [tex]\( f(-1) - f(-2) = 5 - 14 = -9 \)[/tex]
- [tex]\( f(0) - f(-1) = 0 - 5 = -5 \)[/tex]
- [tex]\( f(1) - f(0) = 5 - 0 = 5 \)[/tex]
- [tex]\( f(2) - f(1) = 26 - 5 = 21 \)[/tex]
- For [tex]\(\Delta g\)[/tex] (differences in [tex]\( g(x) \)[/tex]):
- [tex]\( g(-1) - g(-2) = 8 - (-4) = 12 \)[/tex]
- [tex]\( g(0) - g(-1) = 6 - 8 = -2 \)[/tex]
- [tex]\( g(1) - g(0) = 2 - 6 = -4 \)[/tex]
- [tex]\( g(2) - g(1) = -16 - 2 = -18 \)[/tex]
### Step 4: Calculate the mean rate of change
- Mean rate of change for [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Mean rate for } f = \frac{\Delta f_{\text{total}}}{\text{number of intervals}} = \frac{-9 + (-5) + 5 + 21}{4} = \frac{12}{4} = 3 \][/tex]
- Mean rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Mean rate for } g = \frac{\Delta g_{\text{total}}}{\text{number of intervals}} = \frac{12 + (-2) + (-4) + (-18)}{4} = \frac{-12}{4} = -3 \][/tex]
### Step 5: Interpretation of the rates of change
- The mean rate of change for [tex]\( f(x) \)[/tex] over [tex]\([-2, 2]\)[/tex] is [tex]\( 3 \)[/tex], indicating an average increase.
- The mean rate of change for [tex]\( g(x) \)[/tex] over [tex]\([-2, 2]\)[/tex] is [tex]\( -3 \)[/tex], indicating an average decrease.
Therefore, statement A is true: Over the interval [tex]\([-2,2]\)[/tex], function [tex]\( f \)[/tex] is increasing at a faster rate than function [tex]\( g \)[/tex] is decreasing.
### Step 1: Evaluate [tex]\( f(x) \)[/tex] at each given [tex]\( x \)[/tex] value.
Given the values of [tex]\( x \)[/tex] are [tex]\([-2, -1, 0, 1, 2]\)[/tex], we compute [tex]\( f(x) \)[/tex]:
- [tex]\( f(-2) = (-2)^3 + 5(-2)^2 - (-2) = -8 + 20 + 2 = 14 \)[/tex]
- [tex]\( f(-1) = (-1)^3 + 5(-1)^2 - (-1) = -1 + 5 + 1 = 5 \)[/tex]
- [tex]\( f(0) = 0^3 + 5(0)^2 - 0 = 0 \)[/tex]
- [tex]\( f(1) = 1^3 + 5(1)^2 - 1 = 1 + 5 - 1 = 5 \)[/tex]
- [tex]\( f(2) = 2^3 + 5(2)^2 - 2 = 8 + 20 - 2 = 26 \)[/tex]
So, the values of [tex]\( f(x) \)[/tex] at [tex]\([-2, -1, 0, 1, 2]\)[/tex] are [tex]\( [14, 5, 0, 5, 26] \)[/tex].
### Step 2: Use the provided values of [tex]\( g(x) \)[/tex]
From the given table, we have the values of [tex]\( g(x) \)[/tex] at [tex]\([-2, -1, 0, 1, 2]\)[/tex] as [tex]\([-4, 8, 6, 2, -16]\)[/tex].
### Step 3: Calculate the rate of change for each function over the interval [tex]\([-2, 2]\)[/tex]
- For [tex]\(\Delta f\)[/tex] (differences in [tex]\( f(x) \)[/tex]):
- [tex]\( f(-1) - f(-2) = 5 - 14 = -9 \)[/tex]
- [tex]\( f(0) - f(-1) = 0 - 5 = -5 \)[/tex]
- [tex]\( f(1) - f(0) = 5 - 0 = 5 \)[/tex]
- [tex]\( f(2) - f(1) = 26 - 5 = 21 \)[/tex]
- For [tex]\(\Delta g\)[/tex] (differences in [tex]\( g(x) \)[/tex]):
- [tex]\( g(-1) - g(-2) = 8 - (-4) = 12 \)[/tex]
- [tex]\( g(0) - g(-1) = 6 - 8 = -2 \)[/tex]
- [tex]\( g(1) - g(0) = 2 - 6 = -4 \)[/tex]
- [tex]\( g(2) - g(1) = -16 - 2 = -18 \)[/tex]
### Step 4: Calculate the mean rate of change
- Mean rate of change for [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Mean rate for } f = \frac{\Delta f_{\text{total}}}{\text{number of intervals}} = \frac{-9 + (-5) + 5 + 21}{4} = \frac{12}{4} = 3 \][/tex]
- Mean rate of change for [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Mean rate for } g = \frac{\Delta g_{\text{total}}}{\text{number of intervals}} = \frac{12 + (-2) + (-4) + (-18)}{4} = \frac{-12}{4} = -3 \][/tex]
### Step 5: Interpretation of the rates of change
- The mean rate of change for [tex]\( f(x) \)[/tex] over [tex]\([-2, 2]\)[/tex] is [tex]\( 3 \)[/tex], indicating an average increase.
- The mean rate of change for [tex]\( g(x) \)[/tex] over [tex]\([-2, 2]\)[/tex] is [tex]\( -3 \)[/tex], indicating an average decrease.
Therefore, statement A is true: Over the interval [tex]\([-2,2]\)[/tex], function [tex]\( f \)[/tex] is increasing at a faster rate than function [tex]\( g \)[/tex] is decreasing.
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