### Calculus-Based Graphing of a Polynomial FunctionTo understand the behavior of the following polynomial function, you will sketch its graph using calculus principles. This exercise will help you answer the next four questions.\[ y = \frac{x^3}{3} + \frac{x^2}{2} - 6x + 4 \]#### Key Steps:1. **Find the First Derivative ($ y' $)** - The first derivative provides information about the slope of the function, and it helps identify critical points (maximums, minimums, and points of inflection).2. **Find the Second Derivative ($ y'' $)** - The second derivative gives information on the concavity of the function and helps determine the nature of critical points.3. **Determine Critical Points** - Set the first derivative equal to zero to find critical points. These can indicate where the function has local maxima and minima.4. **Analyze Concavity and Points of Inflection** - Use the second derivative to determine intervals where the function is concave up or down. Points where the concavity changes are called points of inflection.5. **Sketch the Graph** - Use the critical points, points of inflection, and the behavior at infinity to sketch the graph accurately.Use this sketch to answer subsequent questions about the function's behavior and properties.---Note: Please proceed with the detailed calculations and graph sketching based on the derivative results for a comprehensive understanding.

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Answered: Sketch a graph of this function using…

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Sketch a graph of this function using calculus and use it to answer the next 4 questions. 3 x y = + - бх + 4 3

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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### Calculus-Based Graphing of a Polynomial Function

To understand the behavior of the following polynomial function, you will sketch its graph using calculus principles. This exercise will help you answer the next four questions.

\[ y = \frac{x^3}{3} + \frac{x^2}{2} - 6x + 4 \]

#### Key Steps:

1. **Find the First Derivative ($ y' $)**
- The first derivative provides information about the slope of the function, and it helps identify critical points (maximums, minimums, and points of inflection).

2. **Find the Second Derivative ($ y'' $)**
- The second derivative gives information on the concavity of the function and helps determine the nature of critical points.

3. **Determine Critical Points**
- Set the first derivative equal to zero to find critical points. These can indicate where the function has local maxima and minima.

4. **Analyze Concavity and Points of Inflection**
- Use the second derivative to determine intervals where the function is concave up or down. Points where the concavity changes are called points of inflection.

5. **Sketch the Graph**
- Use the critical points, points of inflection, and the behavior at infinity to sketch the graph accurately.

Use this sketch to answer subsequent questions about the function's behavior and properties.

---
Note: Please proceed with the detailed calculations and graph sketching based on the derivative results for a comprehensive understanding.

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