Answer :
Certainly! To analyze the function [tex]\( f(x) = -0.25x^4 + x^3 + 1 \)[/tex], we will break it down step-by-step using calculus concepts such as differentiation, critical points, and concavity. Here's the complete solution:
1. First Derivative:
- The first derivative of the function [tex]\( f(x) \)[/tex] is given by:
[tex]\[ f'(x) = -x^3 + 3x^2 \][/tex]
- We use this to find the critical points and to determine where the function is increasing or decreasing.
2. Second Derivative:
- The second derivative of the function [tex]\( f(x) \)[/tex] is given by:
[tex]\[ f''(x) = -3x^2 + 6x \][/tex]
- We use this to find inflection points and to determine concavity.
3. Critical Points:
- Critical points occur where [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 = 0 \][/tex]
- Factoring the equation:
[tex]\[ x^2(-x + 3) = 0 \][/tex]
- So, the critical points are:
[tex]\[ x = 0, \quad x = 3 \][/tex]
4. Increasing and Decreasing Intervals:
- To determine where the function is increasing or decreasing, we analyze the sign of [tex]\( f'(x) \)[/tex]:
- [tex]\( f'(x) > 0 \)[/tex] implies the function is increasing.
- [tex]\( f'(x) < 0 \)[/tex] implies the function is decreasing.
- Solving [tex]\( f'(x) > 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 > 0 \][/tex]
- The function is increasing in the interval:
[tex]\[ (-\infty, 3) \quad \text{excluding} \quad x=0 \][/tex]
- Solving [tex]\( f'(x) < 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 < 0 \][/tex]
- The function is decreasing in the interval:
[tex]\[ (3, \infty) \][/tex]
5. Inflection Points:
- Inflection points occur where [tex]\( f''(x) = 0 \)[/tex]:
[tex]\[ -3x^2 + 6x = 0 \][/tex]
- Factoring the equation:
[tex]\[ -3x(x - 2) = 0 \][/tex]
- So, the inflection points are:
[tex]\[ x = 0, \quad x = 2 \][/tex]
6. Concavity:
- To determine concavity, we analyze the sign of [tex]\( f''(x) \)[/tex]:
- [tex]\( f''(x) > 0 \)[/tex] implies the function is concave up.
- [tex]\( f''(x) < 0 \)[/tex] implies the function is concave down.
- Solving [tex]\( f''(x) > 0 \)[/tex]:
[tex]\[ -3x^2 + 6x > 0 \][/tex]
- The function is concave up in the interval:
[tex]\[ (0, 2) \][/tex]
- Solving [tex]\( f''(x) < 0 \)[/tex]:
[tex]\[ -3x^2 + 6x < 0 \][/tex]
- The function is concave down in the intervals:
[tex]\[ (-\infty, 0) \cup (2, \infty) \][/tex]
7. Relative Extrema:
- Substituting the critical points into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 1, \quad f(3) = 7.75 \][/tex]
- Therefore, the relative extrema are:
[tex]\[ (0, 1) \quad \text{and} \quad (3, 7.75) \][/tex]
8. Inflection Points Values:
- Substituting the inflection points into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 1, \quad f(2) = 5 \][/tex]
- Therefore, the inflection points are:
[tex]\[ (0, 1) \quad \text{and} \quad (2, 5) \][/tex]
9. X-Intercepts (Roots):
- Solving [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -0.25x^4 + x^3 + 1 = 0 \][/tex]
- The x-intercepts are approximately:
[tex]\[ x \approx -0.933, \quad x \approx 4.060 \][/tex]
- Additionally, the complex solutions:
[tex]\[ x \approx 0.436 \pm i0.931 \][/tex]
10. Y-Intercept:
- The y-intercept occurs where [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 1 \][/tex]
- Thus, the y-intercept is:
[tex]\[ (0, 1) \][/tex]
11. Domain:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers:
[tex]\[ (-\infty, \infty) \][/tex]
Graphing:
Here's the summary of key points to include in the graph:
- Critical Points (Relative Extrema): [tex]\((0, 1)\)[/tex] and [tex]\((3, 7.75)\)[/tex]
- Inflection Points: [tex]\((0, 1)\)[/tex] and [tex]\((2, 5)\)[/tex]
- X-Intercepts: Approximately at [tex]\( x \approx -0.933 \)[/tex] and [tex]\( x \approx 4.060 \)[/tex]
- Y-Intercept: [tex]\((0, 1)\)[/tex]
- Increasing Intervals: [tex]\((-∞, 3)\)[/tex] excluding 0
- Decreasing Intervals: [tex]\((3, ∞)\)[/tex]
- Concave Up Intervals: [tex]\((0, 2)\)[/tex]
- Concave Down Intervals: [tex]\((-∞, 0)∪(2, ∞)\)[/tex]
- Domain: All real numbers, [tex]\((-\infty, \infty)\)[/tex]
Make sure to plot these points and label them along with the intervals of increase/decrease and concavity on your graph for a complete visual understanding of the function.
1. First Derivative:
- The first derivative of the function [tex]\( f(x) \)[/tex] is given by:
[tex]\[ f'(x) = -x^3 + 3x^2 \][/tex]
- We use this to find the critical points and to determine where the function is increasing or decreasing.
2. Second Derivative:
- The second derivative of the function [tex]\( f(x) \)[/tex] is given by:
[tex]\[ f''(x) = -3x^2 + 6x \][/tex]
- We use this to find inflection points and to determine concavity.
3. Critical Points:
- Critical points occur where [tex]\( f'(x) = 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 = 0 \][/tex]
- Factoring the equation:
[tex]\[ x^2(-x + 3) = 0 \][/tex]
- So, the critical points are:
[tex]\[ x = 0, \quad x = 3 \][/tex]
4. Increasing and Decreasing Intervals:
- To determine where the function is increasing or decreasing, we analyze the sign of [tex]\( f'(x) \)[/tex]:
- [tex]\( f'(x) > 0 \)[/tex] implies the function is increasing.
- [tex]\( f'(x) < 0 \)[/tex] implies the function is decreasing.
- Solving [tex]\( f'(x) > 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 > 0 \][/tex]
- The function is increasing in the interval:
[tex]\[ (-\infty, 3) \quad \text{excluding} \quad x=0 \][/tex]
- Solving [tex]\( f'(x) < 0 \)[/tex]:
[tex]\[ -x^3 + 3x^2 < 0 \][/tex]
- The function is decreasing in the interval:
[tex]\[ (3, \infty) \][/tex]
5. Inflection Points:
- Inflection points occur where [tex]\( f''(x) = 0 \)[/tex]:
[tex]\[ -3x^2 + 6x = 0 \][/tex]
- Factoring the equation:
[tex]\[ -3x(x - 2) = 0 \][/tex]
- So, the inflection points are:
[tex]\[ x = 0, \quad x = 2 \][/tex]
6. Concavity:
- To determine concavity, we analyze the sign of [tex]\( f''(x) \)[/tex]:
- [tex]\( f''(x) > 0 \)[/tex] implies the function is concave up.
- [tex]\( f''(x) < 0 \)[/tex] implies the function is concave down.
- Solving [tex]\( f''(x) > 0 \)[/tex]:
[tex]\[ -3x^2 + 6x > 0 \][/tex]
- The function is concave up in the interval:
[tex]\[ (0, 2) \][/tex]
- Solving [tex]\( f''(x) < 0 \)[/tex]:
[tex]\[ -3x^2 + 6x < 0 \][/tex]
- The function is concave down in the intervals:
[tex]\[ (-\infty, 0) \cup (2, \infty) \][/tex]
7. Relative Extrema:
- Substituting the critical points into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 1, \quad f(3) = 7.75 \][/tex]
- Therefore, the relative extrema are:
[tex]\[ (0, 1) \quad \text{and} \quad (3, 7.75) \][/tex]
8. Inflection Points Values:
- Substituting the inflection points into [tex]\( f(x) \)[/tex]:
[tex]\[ f(0) = 1, \quad f(2) = 5 \][/tex]
- Therefore, the inflection points are:
[tex]\[ (0, 1) \quad \text{and} \quad (2, 5) \][/tex]
9. X-Intercepts (Roots):
- Solving [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -0.25x^4 + x^3 + 1 = 0 \][/tex]
- The x-intercepts are approximately:
[tex]\[ x \approx -0.933, \quad x \approx 4.060 \][/tex]
- Additionally, the complex solutions:
[tex]\[ x \approx 0.436 \pm i0.931 \][/tex]
10. Y-Intercept:
- The y-intercept occurs where [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 1 \][/tex]
- Thus, the y-intercept is:
[tex]\[ (0, 1) \][/tex]
11. Domain:
- The domain of [tex]\( f(x) \)[/tex] is all real numbers:
[tex]\[ (-\infty, \infty) \][/tex]
Graphing:
Here's the summary of key points to include in the graph:
- Critical Points (Relative Extrema): [tex]\((0, 1)\)[/tex] and [tex]\((3, 7.75)\)[/tex]
- Inflection Points: [tex]\((0, 1)\)[/tex] and [tex]\((2, 5)\)[/tex]
- X-Intercepts: Approximately at [tex]\( x \approx -0.933 \)[/tex] and [tex]\( x \approx 4.060 \)[/tex]
- Y-Intercept: [tex]\((0, 1)\)[/tex]
- Increasing Intervals: [tex]\((-∞, 3)\)[/tex] excluding 0
- Decreasing Intervals: [tex]\((3, ∞)\)[/tex]
- Concave Up Intervals: [tex]\((0, 2)\)[/tex]
- Concave Down Intervals: [tex]\((-∞, 0)∪(2, ∞)\)[/tex]
- Domain: All real numbers, [tex]\((-\infty, \infty)\)[/tex]
Make sure to plot these points and label them along with the intervals of increase/decrease and concavity on your graph for a complete visual understanding of the function.
