Answer :
Let's analyze both functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] over the interval [tex]\([-2, 2]\)[/tex].
Given:
[tex]\[ f(x) = x^3 + 5x^2 - x \][/tex]
[tex]\[ g(x) \ \text{for specific values of} \ x \][/tex]
We need to find the values of [tex]\( f(x) \)[/tex] at specific points and then determine the rates of change for both functions over the interval. We will calculate these in steps:
1. Calculate the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2, -1, 0, 1, 2 \)[/tex]:
- [tex]\( f(-2) = (-2)^3 + 5(-2)^2 - (-2) \)[/tex]
- [tex]\( f(-1) = (-1)^3 + 5(-1)^2 - (-1) \)[/tex]
- [tex]\( f(0) = 0^3 + 5(0)^2 - 0 \)[/tex]
- [tex]\( f(1) = 1^3 + 5(1)^2 - 1 \)[/tex]
- [tex]\( f(2) = 2^3 + 5(2)^2 - 2 \)[/tex]
By evaluating these,
[tex]\[ f(-2) = 14, \quad f(-1) = 5, \quad f(0) = 0, \quad f(1) = 5, \quad f(2) = 26 \][/tex]
Hence, the values of [tex]\( f \)[/tex] over the interval are: [tex]\([14, 5, 0, 5, 26]\)[/tex].
2. Given values of [tex]\( g(x) \)[/tex]:
[tex]\[ g(-2) = -4, \quad g(-1) = 8, \quad g(0) = 6, \quad g(1) = 2, \quad g(2) = -16 \][/tex]
Hence, the values of [tex]\( g \)[/tex] over the interval are: [tex]\([-4, 8, 6, 2, -16]\)[/tex].
3. Calculate the rate of change for [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
For [tex]\( f \)[/tex]:
[tex]\[ \Delta f_{-2 \to -1} = 5 - 14 = -9, \quad \Delta f_{-1 \to 0} = 0 - 5 = -5, \quad \Delta f_{0 \to 1} = 5 - 0 = 5, \quad \Delta f_{1 \to 2} = 26 - 5 = 21 \][/tex]
The rate of change of [tex]\( f \)[/tex] is [tex]\([-9, -5, 5, 21]\)[/tex].
For [tex]\( g \)[/tex]:
[tex]\[ \Delta g_{-2 \to -1} = 8 - (-4) = 12, \quad \Delta g_{-1 \to 0} = 6 - 8 = -2, \quad \Delta g_{0 \to 1} = 2 - 6 = -4, \quad \Delta g_{1 \to 2} = -16 - 2 = -18 \][/tex]
The rate of change of [tex]\( g \)[/tex] is [tex]\([12, -2, -4, -18]\)[/tex].
4. Determine the true statement:
Considering the rates of change calculated, none of the options:
- Rates of change of [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not consistently the same.
- [tex]\( f \)[/tex] is not increasing; it has both increasing and decreasing segments, thus cannot be compared directly to [tex]\( g \)[/tex] consistently.
- [tex]\( f \)[/tex] and [tex]\( g \)[/tex]'s rates of change in magnitude and direction do not match consistently.
Therefore, the correct answer based on examining these calculations:
[tex]\[ \text{D. Over the interval } [-2,2], \text{ function } f \text{ is decreasing at a faster rate than function } g \text{ is increasing.} \][/tex]
This is closest since [tex]\( f \)[/tex] starts decreasing more rapidly before increasing and has segments that vary in rate compared to [tex]\( g \)[/tex]. However, if only these options are present, we default to option [tex]\( \text{D} \)[/tex].
Given:
[tex]\[ f(x) = x^3 + 5x^2 - x \][/tex]
[tex]\[ g(x) \ \text{for specific values of} \ x \][/tex]
We need to find the values of [tex]\( f(x) \)[/tex] at specific points and then determine the rates of change for both functions over the interval. We will calculate these in steps:
1. Calculate the values of [tex]\( f(x) \)[/tex] at [tex]\( x = -2, -1, 0, 1, 2 \)[/tex]:
- [tex]\( f(-2) = (-2)^3 + 5(-2)^2 - (-2) \)[/tex]
- [tex]\( f(-1) = (-1)^3 + 5(-1)^2 - (-1) \)[/tex]
- [tex]\( f(0) = 0^3 + 5(0)^2 - 0 \)[/tex]
- [tex]\( f(1) = 1^3 + 5(1)^2 - 1 \)[/tex]
- [tex]\( f(2) = 2^3 + 5(2)^2 - 2 \)[/tex]
By evaluating these,
[tex]\[ f(-2) = 14, \quad f(-1) = 5, \quad f(0) = 0, \quad f(1) = 5, \quad f(2) = 26 \][/tex]
Hence, the values of [tex]\( f \)[/tex] over the interval are: [tex]\([14, 5, 0, 5, 26]\)[/tex].
2. Given values of [tex]\( g(x) \)[/tex]:
[tex]\[ g(-2) = -4, \quad g(-1) = 8, \quad g(0) = 6, \quad g(1) = 2, \quad g(2) = -16 \][/tex]
Hence, the values of [tex]\( g \)[/tex] over the interval are: [tex]\([-4, 8, 6, 2, -16]\)[/tex].
3. Calculate the rate of change for [tex]\( f \)[/tex] and [tex]\( g \)[/tex]:
For [tex]\( f \)[/tex]:
[tex]\[ \Delta f_{-2 \to -1} = 5 - 14 = -9, \quad \Delta f_{-1 \to 0} = 0 - 5 = -5, \quad \Delta f_{0 \to 1} = 5 - 0 = 5, \quad \Delta f_{1 \to 2} = 26 - 5 = 21 \][/tex]
The rate of change of [tex]\( f \)[/tex] is [tex]\([-9, -5, 5, 21]\)[/tex].
For [tex]\( g \)[/tex]:
[tex]\[ \Delta g_{-2 \to -1} = 8 - (-4) = 12, \quad \Delta g_{-1 \to 0} = 6 - 8 = -2, \quad \Delta g_{0 \to 1} = 2 - 6 = -4, \quad \Delta g_{1 \to 2} = -16 - 2 = -18 \][/tex]
The rate of change of [tex]\( g \)[/tex] is [tex]\([12, -2, -4, -18]\)[/tex].
4. Determine the true statement:
Considering the rates of change calculated, none of the options:
- Rates of change of [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are not consistently the same.
- [tex]\( f \)[/tex] is not increasing; it has both increasing and decreasing segments, thus cannot be compared directly to [tex]\( g \)[/tex] consistently.
- [tex]\( f \)[/tex] and [tex]\( g \)[/tex]'s rates of change in magnitude and direction do not match consistently.
Therefore, the correct answer based on examining these calculations:
[tex]\[ \text{D. Over the interval } [-2,2], \text{ function } f \text{ is decreasing at a faster rate than function } g \text{ is increasing.} \][/tex]
This is closest since [tex]\( f \)[/tex] starts decreasing more rapidly before increasing and has segments that vary in rate compared to [tex]\( g \)[/tex]. However, if only these options are present, we default to option [tex]\( \text{D} \)[/tex].
