Answer :
Let's analyze the given statements one at a time and compare them against the results we derived from the functions and their derivatives.
### Statement a: [tex]\(F(x)=x^3-4x+100\)[/tex] and [tex]\(G(x)=x^3-4x-100\)[/tex] are antiderivatives of the same function.
#### Options for Statement a:
1. Option A. The statement is true. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3x^2-4\)[/tex].
2. Option B. The statement is true. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3x-4\)[/tex].
3. Option C. The statement is false. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=\frac{x^4}{4}-2 x^2+100 x\)[/tex] and [tex]\(G^{\prime}(x)=\frac{x^4}{4}-2 x^2-100 x\)[/tex].
4. Option D. The statement is false. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3 x^2-4\)[/tex].
Explanation:
When we differentiate both functions:
[tex]\[ F(x) = x^3 - 4x + 100 \][/tex]
[tex]\[ G(x) = x^3 - 4x - 100 \][/tex]
Taking their derivatives:
[tex]\[ F^{\prime}(x) = 3x^2 - 4 \][/tex]
[tex]\[ G^{\prime}(x) = 3x^2 - 4 \][/tex]
Since [tex]\(F^{\prime}(x)\)[/tex] and [tex]\(G^{\prime}(x)\)[/tex] are equal to [tex]\(3x^2 - 4\)[/tex], the correct option must recognize this equality properly.
Correct Option:
- Option A is true and accurately states that [tex]\(F^{\prime}(x) = G^{\prime}(x) = 3x^2 - 4\)[/tex].
### Statement b: If [tex]\(F^{\prime}(x) = f(x)\)[/tex], then [tex]\(f\)[/tex] is the antiderivative of [tex]\(F\)[/tex].
#### Options for Statement b:
1. Option A. The statement is false because the antiderivative of [tex]\(F\)[/tex] may vary depending on the constant of integration.
2. Option B. The statement is true because a function [tex]\(f\)[/tex] is an antiderivative of [tex]\(f\)[/tex] on an interval [tex]\(I\)[/tex] provided [tex]\(F^{\prime}(x) = f(x)\)[/tex], for all [tex]\(x\)[/tex] in [tex]\(I\)[/tex].
3. Option C. The statement is true. Let [tex]\(F^{\prime}(x) = f(x) = x^2\)[/tex]. Since [tex]\(\int x^2 \, dx = 2x\)[/tex] for all [tex]\(x\)[/tex], [tex]\(f\)[/tex] is the antiderivative of [tex]\(F\)[/tex].
4. Option D. The statement is false, because a function [tex]\(F\)[/tex] is an antiderivative of [tex]\(f\)[/tex] on an interval [tex]\(I\)[/tex] provided [tex]\(F^{\prime}(x) = f(x)\)[/tex], for all [tex]\(x\)[/tex] in [tex]\(I\)[/tex].
Explanation:
For [tex]\(F^{\prime}(x) = f(x)\)[/tex], [tex]\(f(x)\)[/tex] is the derivative of [tex]\(F(x)\)[/tex]. This implies that [tex]\(f\)[/tex] is the function such that integrating it over [tex]\(x\)[/tex] gives the function [tex]\(F(x)\)[/tex] plus a constant. Thus [tex]\(f\)[/tex] is indeed the derivative (or the antiderivative) that represents the rate of change of [tex]\(F(x)\)[/tex].
Correct Option:
- Option B is accurate and states the role of [tex]\(f(x)\)[/tex] as the antiderivative required for [tex]\(F(x)\)[/tex].
### Conclusion:
- Statement a: Option A is the correct answer.
- Statement b: Option B is the correct answer.
These conclusions match the derived results and fulfill the conditions given in the question.
### Statement a: [tex]\(F(x)=x^3-4x+100\)[/tex] and [tex]\(G(x)=x^3-4x-100\)[/tex] are antiderivatives of the same function.
#### Options for Statement a:
1. Option A. The statement is true. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3x^2-4\)[/tex].
2. Option B. The statement is true. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3x-4\)[/tex].
3. Option C. The statement is false. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=\frac{x^4}{4}-2 x^2+100 x\)[/tex] and [tex]\(G^{\prime}(x)=\frac{x^4}{4}-2 x^2-100 x\)[/tex].
4. Option D. The statement is false. For the given [tex]\(F(x)\)[/tex] and [tex]\(G(x)\)[/tex], [tex]\(F^{\prime}(x)=G^{\prime}(x)=3 x^2-4\)[/tex].
Explanation:
When we differentiate both functions:
[tex]\[ F(x) = x^3 - 4x + 100 \][/tex]
[tex]\[ G(x) = x^3 - 4x - 100 \][/tex]
Taking their derivatives:
[tex]\[ F^{\prime}(x) = 3x^2 - 4 \][/tex]
[tex]\[ G^{\prime}(x) = 3x^2 - 4 \][/tex]
Since [tex]\(F^{\prime}(x)\)[/tex] and [tex]\(G^{\prime}(x)\)[/tex] are equal to [tex]\(3x^2 - 4\)[/tex], the correct option must recognize this equality properly.
Correct Option:
- Option A is true and accurately states that [tex]\(F^{\prime}(x) = G^{\prime}(x) = 3x^2 - 4\)[/tex].
### Statement b: If [tex]\(F^{\prime}(x) = f(x)\)[/tex], then [tex]\(f\)[/tex] is the antiderivative of [tex]\(F\)[/tex].
#### Options for Statement b:
1. Option A. The statement is false because the antiderivative of [tex]\(F\)[/tex] may vary depending on the constant of integration.
2. Option B. The statement is true because a function [tex]\(f\)[/tex] is an antiderivative of [tex]\(f\)[/tex] on an interval [tex]\(I\)[/tex] provided [tex]\(F^{\prime}(x) = f(x)\)[/tex], for all [tex]\(x\)[/tex] in [tex]\(I\)[/tex].
3. Option C. The statement is true. Let [tex]\(F^{\prime}(x) = f(x) = x^2\)[/tex]. Since [tex]\(\int x^2 \, dx = 2x\)[/tex] for all [tex]\(x\)[/tex], [tex]\(f\)[/tex] is the antiderivative of [tex]\(F\)[/tex].
4. Option D. The statement is false, because a function [tex]\(F\)[/tex] is an antiderivative of [tex]\(f\)[/tex] on an interval [tex]\(I\)[/tex] provided [tex]\(F^{\prime}(x) = f(x)\)[/tex], for all [tex]\(x\)[/tex] in [tex]\(I\)[/tex].
Explanation:
For [tex]\(F^{\prime}(x) = f(x)\)[/tex], [tex]\(f(x)\)[/tex] is the derivative of [tex]\(F(x)\)[/tex]. This implies that [tex]\(f\)[/tex] is the function such that integrating it over [tex]\(x\)[/tex] gives the function [tex]\(F(x)\)[/tex] plus a constant. Thus [tex]\(f\)[/tex] is indeed the derivative (or the antiderivative) that represents the rate of change of [tex]\(F(x)\)[/tex].
Correct Option:
- Option B is accurate and states the role of [tex]\(f(x)\)[/tex] as the antiderivative required for [tex]\(F(x)\)[/tex].
### Conclusion:
- Statement a: Option A is the correct answer.
- Statement b: Option B is the correct answer.
These conclusions match the derived results and fulfill the conditions given in the question.
