You use 'INF' for oo and '-INF' for -∞o. And use 'U' for the union symbol. Enter DNE if an answer does not exist. f(x) = 2x² - 2x + 4 a) Determine the intervals on which f is concave up and concave down. f is concave up on: f is concave down on: b) Based on your answer to part (a), determine the inflection points of f. Each point should be entered as an ordered pair (that is, in the form (x, y)). (Separate multiple answers by commas.) c) Find the critical numbers of f and use the Second Derivative Test, when possible, to determine the relative extrema. List only the x-coordinates. Relative maxima at: Relative minima at: (Separate multiple answers by commas.) (Separate multiple answers by commas.)

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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On this educational website, we will delve into the analysis of the function \( f(x) = -2x^2 - 2x + 4 \). Follow the instructions below to complete the tasks associated with understanding the concavity, inflection points, and relative extrema of the function.

### Instructions:

1. **You use 'INF' for \(\infty\) and '-INF' for \(-\infty\).**
2. **And use 'U' for the union symbol.**
3. **Enter DNE if an answer does not exist.**

### Function Analysis:
\[ f(x) = -2x^2 - 2x + 4 \]

#### a) Determine the intervals on which \( f \) is concave up and concave down.

- \( f \) is concave up on: \(\_\_\_\_\_\_\_\_\_\_\_\)
- \( f \) is concave down on: \(\_\_\_\_\_\_\_\_\_\_\_\)

#### b) Based on your answer to part (a), determine the inflection points of \( f \). Each point should be entered as an **ordered pair** (that is, in the form (x, y)).
\[ ( \_\_\_\_\_, \_\_\_\_\_ ) \]
*(Separate multiple answers by commas.)*

#### c) Find the critical numbers of \( f \) and use the Second Derivative Test, when possible, to determine the relative extrema. List only the \( x \)-coordinates.

- **Relative maxima at:**
\[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ]
*(Separate multiple answers by commas.)*

- **Relative minima at:**
\[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ]
*(Separate multiple answers by commas.)*

#### Explanation for Educators:
This series of questions is targeted at assessing a student's understanding of concavity, inflection points, and relative extrema using a given quadratic function. The students are expected to use calculus principles, such as the second derivative test, to find intervals of concavity and critical points effectively.

Feel free to reach out if you have any questions or need additional clarifications on this topic.
Transcribed Image Text:On this educational website, we will delve into the analysis of the function \( f(x) = -2x^2 - 2x + 4 \). Follow the instructions below to complete the tasks associated with understanding the concavity, inflection points, and relative extrema of the function. ### Instructions: 1. **You use 'INF' for \(\infty\) and '-INF' for \(-\infty\).** 2. **And use 'U' for the union symbol.** 3. **Enter DNE if an answer does not exist.** ### Function Analysis: \[ f(x) = -2x^2 - 2x + 4 \] #### a) Determine the intervals on which \( f \) is concave up and concave down. - \( f \) is concave up on: \(\_\_\_\_\_\_\_\_\_\_\_\) - \( f \) is concave down on: \(\_\_\_\_\_\_\_\_\_\_\_\) #### b) Based on your answer to part (a), determine the inflection points of \( f \). Each point should be entered as an **ordered pair** (that is, in the form (x, y)). \[ ( \_\_\_\_\_, \_\_\_\_\_ ) \] *(Separate multiple answers by commas.)* #### c) Find the critical numbers of \( f \) and use the Second Derivative Test, when possible, to determine the relative extrema. List only the \( x \)-coordinates. - **Relative maxima at:** \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ] *(Separate multiple answers by commas.)* - **Relative minima at:** \[ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\ ] *(Separate multiple answers by commas.)* #### Explanation for Educators: This series of questions is targeted at assessing a student's understanding of concavity, inflection points, and relative extrema using a given quadratic function. The students are expected to use calculus principles, such as the second derivative test, to find intervals of concavity and critical points effectively. Feel free to reach out if you have any questions or need additional clarifications on this topic.
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