Answer :
Let's analyze the polynomial function [tex]\(F(x) = 2x^3 - 2x^2 + 1\)[/tex]:
1. Finding the First Derivative:
To locate the relative extrema, we first take the derivative of [tex]\(F(x)\)[/tex]:
[tex]\[ F'(x) = \frac{d}{dx}(2x^3 - 2x^2 + 1) = 6x^2 - 4x \][/tex]
2. Setting the First Derivative to Zero:
To find the critical points, we solve for [tex]\(x\)[/tex] when [tex]\(F'(x) = 0\)[/tex]:
[tex]\[ 6x^2 - 4x = 0 \][/tex]
Factor the equation:
[tex]\[ 2x(3x - 2) = 0 \][/tex]
So, the critical points are:
[tex]\[ x = 0 \quad \text{and} \quad x = \frac{2}{3} \][/tex]
3. Finding the Second Derivative:
To determine the nature of each critical point, we take the second derivative of [tex]\(F(x)\)[/tex]:
[tex]\[ F''(x) = \frac{d}{dx}(6x^2 - 4x) = 12x - 4 \][/tex]
4. Evaluating the Second Derivative at the Critical Points:
- At [tex]\(x = 0\)[/tex]:
[tex]\[ F''(0) = 12 \cdot 0 - 4 = -4 \implies \text{relative maximum} \quad (\text{since}\ F''(0) < 0) \][/tex]
- At [tex]\(x = \frac{2}{3}\)[/tex]:
[tex]\[ F''\left(\frac{2}{3}\right) = 12 \cdot \frac{2}{3} - 4 = 8 \implies \text{relative minimum} \quad (\text{since}\ F''\left(\frac{2}{3}\right) > 0) \][/tex]
5. Analyzing the Limits as [tex]\(x\)[/tex] Approaches Infinity and Negative Infinity:
- As [tex]\(x \rightarrow \infty\)[/tex], the dominant term is [tex]\(2x^3\)[/tex], so:
[tex]\[ F(x) \rightarrow \infty \][/tex]
- As [tex]\(x \rightarrow -\infty\)[/tex], considering [tex]\(2x^3\)[/tex] again:
[tex]\[ F(x) \rightarrow -\infty \][/tex]
Summary of the Analysis:
- The function [tex]\(F(x)\)[/tex] has 1 relative minimum and 1 relative maximum.
- As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(F(x) \rightarrow \infty\)[/tex].
- As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(F(x) \rightarrow -\infty\)[/tex].
Correct Statement:
A. [tex]\(F(x)\)[/tex] has 1 relative minimum and 1 relative maximum, and as [tex]\(x \rightarrow \infty\)[/tex], [tex]\(F(x) \rightarrow \infty\)[/tex], and as [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(F(x) \rightarrow -\infty\)[/tex].
Thus, the correct options are:
- Option A: True
- Option B: False
- Option C: False
- Option D: True
1. Finding the First Derivative:
To locate the relative extrema, we first take the derivative of [tex]\(F(x)\)[/tex]:
[tex]\[ F'(x) = \frac{d}{dx}(2x^3 - 2x^2 + 1) = 6x^2 - 4x \][/tex]
2. Setting the First Derivative to Zero:
To find the critical points, we solve for [tex]\(x\)[/tex] when [tex]\(F'(x) = 0\)[/tex]:
[tex]\[ 6x^2 - 4x = 0 \][/tex]
Factor the equation:
[tex]\[ 2x(3x - 2) = 0 \][/tex]
So, the critical points are:
[tex]\[ x = 0 \quad \text{and} \quad x = \frac{2}{3} \][/tex]
3. Finding the Second Derivative:
To determine the nature of each critical point, we take the second derivative of [tex]\(F(x)\)[/tex]:
[tex]\[ F''(x) = \frac{d}{dx}(6x^2 - 4x) = 12x - 4 \][/tex]
4. Evaluating the Second Derivative at the Critical Points:
- At [tex]\(x = 0\)[/tex]:
[tex]\[ F''(0) = 12 \cdot 0 - 4 = -4 \implies \text{relative maximum} \quad (\text{since}\ F''(0) < 0) \][/tex]
- At [tex]\(x = \frac{2}{3}\)[/tex]:
[tex]\[ F''\left(\frac{2}{3}\right) = 12 \cdot \frac{2}{3} - 4 = 8 \implies \text{relative minimum} \quad (\text{since}\ F''\left(\frac{2}{3}\right) > 0) \][/tex]
5. Analyzing the Limits as [tex]\(x\)[/tex] Approaches Infinity and Negative Infinity:
- As [tex]\(x \rightarrow \infty\)[/tex], the dominant term is [tex]\(2x^3\)[/tex], so:
[tex]\[ F(x) \rightarrow \infty \][/tex]
- As [tex]\(x \rightarrow -\infty\)[/tex], considering [tex]\(2x^3\)[/tex] again:
[tex]\[ F(x) \rightarrow -\infty \][/tex]
Summary of the Analysis:
- The function [tex]\(F(x)\)[/tex] has 1 relative minimum and 1 relative maximum.
- As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(F(x) \rightarrow \infty\)[/tex].
- As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(F(x) \rightarrow -\infty\)[/tex].
Correct Statement:
A. [tex]\(F(x)\)[/tex] has 1 relative minimum and 1 relative maximum, and as [tex]\(x \rightarrow \infty\)[/tex], [tex]\(F(x) \rightarrow \infty\)[/tex], and as [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(F(x) \rightarrow -\infty\)[/tex].
Thus, the correct options are:
- Option A: True
- Option B: False
- Option C: False
- Option D: True
