Answer :
Alright, let's analyze functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] over the interval [tex]\([-2, 2]\)[/tex] and determine which statement is true.
### Given Data:
- [tex]\( f(x) = x^3 + 5x^2 - x \)[/tex]
- Values of [tex]\( g(x) \)[/tex] at specific [tex]\( x \)[/tex]-values:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & -4 & 8 & 6 & 2 & -16 & -84 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
#### Calculation of [tex]\( f(x) \)[/tex] at Given Points:
First, let's find the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2, -1, 0, 1, 2 \)[/tex].
[tex]\[ \begin{aligned} f(-2) &= (-2)^3 + 5(-2)^2 - (-2) = -8 + 20 + 2 = 14, \\ f(-1) &= (-1)^3 + 5(-1)^2 - (-1) = -1 + 5 + 1 = 5, \\ f(0) &= 0^3 + 5(0)^2 - 0 = 0, \\ f(1) &= 1^3 + 5(1)^2 - 1 = 1 + 5 - 1 = 5, \\ f(2) &= 2^3 + 5(2)^2 - 2 = 8 + 20 - 2 = 26. \end{aligned} \][/tex]
So the [tex]\( f(x) \)[/tex] values at given points are:
[tex]\[ f(-2) = 14, \; f(-1) = 5, \; f(0) = 0, \; f(1) = 5, \; f(2) = 26. \][/tex]
#### Calculation of Rates of Change for [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{aligned} \text{Rate of change from } -2 \text{ to } -1 &: f(-1) - f(-2) = 5 - 14 = -9, \\ \text{Rate of change from } -1 \text{ to } 0 &: f(0) - f(-1) = 0 - 5 = -5, \\ \text{Rate of change from } 0 \text{ to } 1 &: f(1) - f(0) = 5 - 0 = 5, \\ \text{Rate of change from } 1 \text{ to } 2 &: f(2) - f(1) = 26 - 5 = 21. \end{aligned} \][/tex]
Thus, the rates of change for [tex]\( f(x) \)[/tex] are [tex]\([-9, -5, 5, 21]\)[/tex].
#### Calculation of Rates of Change for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{aligned} \text{Rate of change from } -2 \text{ to } -1 &: g(-1) - g(-2) = 8 - (-4) = 12, \\ \text{Rate of change from } -1 \text{ to } 0 &: g(0) - g(-1) = 6 - 8 = -2, \\ \text{Rate of change from } 0 \text{ to } 1 &: g(1) - g(0) = 2 - 6 = -4, \\ \text{Rate of change from } 1 \text{ to } 2 &: g(2) - g(1) = -16 - 2 = -18. \end{aligned} \][/tex]
Thus, the rates of change for [tex]\( g(x) \)[/tex] are [tex]\([12, -2, -4, -18]\)[/tex].
### Analysis of Statements:
#### Statement A: [tex]\( f(x) \)[/tex] is decreasing faster than [tex]\( g(x) \)[/tex] is increasing.
Based on the rates of change, [tex]\( f(x) \)[/tex] has negative rates at [tex]\([-2, -1]\)[/tex] and [tex]\([-1, 0]\)[/tex], but [tex]\( g(x) \)[/tex] is decreasing (negative rates) more often than increasing (positive rate only at [tex]\([-2, -1]\)[/tex]). Therefore, this statement is false due to inconsistent trends within the interval.
#### Statement B: [tex]\( f(x) \)[/tex] is increasing at the same rate that [tex]\( g(x) \)[/tex] is decreasing.
We see that the average rate of change for [tex]\( f(x) \)[/tex] (taking the sum of the rates and dividing by the number of intervals) matches the average rate of change for [tex]\( g(x) \)[/tex] in absolute terms but opposite in sign. This makes the statement true.
#### Statement C: Both functions are decreasing at the same rate.
This is incorrect because both functions are not always decreasing (some rates of change for [tex]\( f(x) \)[/tex] are positive).
#### Statement D: [tex]\( f(x) \)[/tex] is increasing faster than [tex]\( g(x) \)[/tex] is decreasing.
This is also false based on the calculated rates of change because [tex]\( f(x) \)[/tex] does not consistently increase faster than [tex]\( g(x) \)[/tex] decreases.
Hence, the correct answer is B. Over the interval [tex]\([-2, 2]\)[/tex], function [tex]\( f \)[/tex] is increasing at the same rate that function [tex]\( g \)[/tex] is decreasing.
### Given Data:
- [tex]\( f(x) = x^3 + 5x^2 - x \)[/tex]
- Values of [tex]\( g(x) \)[/tex] at specific [tex]\( x \)[/tex]-values:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline g(x) & -4 & 8 & 6 & 2 & -16 & -84 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
#### Calculation of [tex]\( f(x) \)[/tex] at Given Points:
First, let's find the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2, -1, 0, 1, 2 \)[/tex].
[tex]\[ \begin{aligned} f(-2) &= (-2)^3 + 5(-2)^2 - (-2) = -8 + 20 + 2 = 14, \\ f(-1) &= (-1)^3 + 5(-1)^2 - (-1) = -1 + 5 + 1 = 5, \\ f(0) &= 0^3 + 5(0)^2 - 0 = 0, \\ f(1) &= 1^3 + 5(1)^2 - 1 = 1 + 5 - 1 = 5, \\ f(2) &= 2^3 + 5(2)^2 - 2 = 8 + 20 - 2 = 26. \end{aligned} \][/tex]
So the [tex]\( f(x) \)[/tex] values at given points are:
[tex]\[ f(-2) = 14, \; f(-1) = 5, \; f(0) = 0, \; f(1) = 5, \; f(2) = 26. \][/tex]
#### Calculation of Rates of Change for [tex]\( f(x) \)[/tex]:
[tex]\[ \begin{aligned} \text{Rate of change from } -2 \text{ to } -1 &: f(-1) - f(-2) = 5 - 14 = -9, \\ \text{Rate of change from } -1 \text{ to } 0 &: f(0) - f(-1) = 0 - 5 = -5, \\ \text{Rate of change from } 0 \text{ to } 1 &: f(1) - f(0) = 5 - 0 = 5, \\ \text{Rate of change from } 1 \text{ to } 2 &: f(2) - f(1) = 26 - 5 = 21. \end{aligned} \][/tex]
Thus, the rates of change for [tex]\( f(x) \)[/tex] are [tex]\([-9, -5, 5, 21]\)[/tex].
#### Calculation of Rates of Change for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{aligned} \text{Rate of change from } -2 \text{ to } -1 &: g(-1) - g(-2) = 8 - (-4) = 12, \\ \text{Rate of change from } -1 \text{ to } 0 &: g(0) - g(-1) = 6 - 8 = -2, \\ \text{Rate of change from } 0 \text{ to } 1 &: g(1) - g(0) = 2 - 6 = -4, \\ \text{Rate of change from } 1 \text{ to } 2 &: g(2) - g(1) = -16 - 2 = -18. \end{aligned} \][/tex]
Thus, the rates of change for [tex]\( g(x) \)[/tex] are [tex]\([12, -2, -4, -18]\)[/tex].
### Analysis of Statements:
#### Statement A: [tex]\( f(x) \)[/tex] is decreasing faster than [tex]\( g(x) \)[/tex] is increasing.
Based on the rates of change, [tex]\( f(x) \)[/tex] has negative rates at [tex]\([-2, -1]\)[/tex] and [tex]\([-1, 0]\)[/tex], but [tex]\( g(x) \)[/tex] is decreasing (negative rates) more often than increasing (positive rate only at [tex]\([-2, -1]\)[/tex]). Therefore, this statement is false due to inconsistent trends within the interval.
#### Statement B: [tex]\( f(x) \)[/tex] is increasing at the same rate that [tex]\( g(x) \)[/tex] is decreasing.
We see that the average rate of change for [tex]\( f(x) \)[/tex] (taking the sum of the rates and dividing by the number of intervals) matches the average rate of change for [tex]\( g(x) \)[/tex] in absolute terms but opposite in sign. This makes the statement true.
#### Statement C: Both functions are decreasing at the same rate.
This is incorrect because both functions are not always decreasing (some rates of change for [tex]\( f(x) \)[/tex] are positive).
#### Statement D: [tex]\( f(x) \)[/tex] is increasing faster than [tex]\( g(x) \)[/tex] is decreasing.
This is also false based on the calculated rates of change because [tex]\( f(x) \)[/tex] does not consistently increase faster than [tex]\( g(x) \)[/tex] decreases.
Hence, the correct answer is B. Over the interval [tex]\([-2, 2]\)[/tex], function [tex]\( f \)[/tex] is increasing at the same rate that function [tex]\( g \)[/tex] is decreasing.
