just need parts D, E, and F that are in red and the inflection point(s). ### Analysis of the Function \( f(x) = \frac{x^3}{x^2 - 12} \)**(a) Domain:**- The domain of \( f \) is \( (-\infty, -2\sqrt{3}) \cup (-2\sqrt{3}, 2\sqrt{3}) \cup (2\sqrt{3}, \infty) \).**(b) Asymptotes:**- **Vertical Asymptotes:** The function \( y = f(x) \) has vertical asymptotes at \( x = 2\sqrt{3} \) and \( x = -2\sqrt{3} \).- **Horizontal Asymptotes:** The function does not have horizontal asymptotes.- **Slant Asymptotes:** The function \( y = f(x) \) has a slant asymptote at \( y = x \).**(c) Symmetry:**- The function \( y = f(x) \) is an even function.**(d) Increasing/Decreasing:**- \( f \) is increasing for \( x \in (-\infty, -6) \cup (6, \infty) \).- \( f \) is not decreasing for any \( x \).**(e) Critical Point Classification:**- \( f \) has a local maximum at \( x = -6 \).- \( f \) has a local minimum at \( x = 6 \).- There are no critical points that are neither local minima nor maxima.**(f) Concavity:**- The function is not concave up for any \( x \).- The function is not concave down for any \( x \).**(g) Inflection Points:**- There are no inflection points. This analysis provides a comprehensive understanding of the behavior of the given function, including its asymptotic behavior, symmetry, and critical points.

Answered: Image /qna-images/answer/1fecc42d-cf85-469f-88b4-6a45b6f7ff04.jpg

x3 Let f(x) x2 12 (a) Domain: The domain of f is (-inf, -2sqrt(3))U(-2sqrt(3),2:E (b) Asymptotes: y = f(x) has vertical asymptote(s) at x = 2sqrt(3), -2sqrt(3) Σ y = f(x) has horizontal asymptote(s) at y = NONE Σ f(x) has slant asymptote(s) at y = Σ Y = || (c) Symmetry: y = f(x) is an even function (d) Increasing / Decreasing: f is increasing for x E (-inf,-6)U(6,inf) Σ f is decreasing for x E NONE Σ (e) Critical Point Classification: f has local maximums at x = -6 Σ f has local minimums at x = 6. Σ f has critical points that are neither local mins nor maxes at x = NONE Σ (f) Concavity: f is concave up for x E NONE Σ f is concave down for x E NONE Σ (g) Inflection Points:

x3 Let f(x) x2 12 (a) Domain: The domain of f is (-inf, -2sqrt(3))U(-2sqrt(3),2:E (b) Asymptotes: y = f(x) has vertical asymptote(s) at x = 2sqrt(3), -2sqrt(3) Σ y = f(x) has horizontal asymptote(s) at y = NONE Σ f(x) has slant asymptote(s) at y = Σ Y = || (c) Symmetry: y = f(x) is an even function (d) Increasing / Decreasing: f is increasing for x E (-inf,-6)U(6,inf) Σ f is decreasing for x E NONE Σ (e) Critical Point Classification: f has local maximums at x = -6 Σ f has local minimums at x = 6. Σ f has critical points that are neither local mins nor maxes at x = NONE Σ (f) Concavity: f is concave up for x E NONE Σ f is concave down for x E NONE Σ (g) Inflection Points:

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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Question

just need parts D, E, and F that are in red and the inflection point(s).

### Analysis of the Function \( f(x) = \frac{x^3}{x^2 - 12} \)

**(a) Domain:**
- The domain of \( f \) is \( (-\infty, -2\sqrt{3}) \cup (-2\sqrt{3}, 2\sqrt{3}) \cup (2\sqrt{3}, \infty) \).

**(b) Asymptotes:**
- **Vertical Asymptotes:** The function \( y = f(x) \) has vertical asymptotes at \( x = 2\sqrt{3} \) and \( x = -2\sqrt{3} \).
- **Horizontal Asymptotes:** The function does not have horizontal asymptotes.
- **Slant Asymptotes:** The function \( y = f(x) \) has a slant asymptote at \( y = x \).

**(c) Symmetry:**
- The function \( y = f(x) \) is an even function.

**(d) Increasing/Decreasing:**
- \( f \) is increasing for \( x \in (-\infty, -6) \cup (6, \infty) \).
- \( f \) is not decreasing for any \( x \).

**(e) Critical Point Classification:**
- \( f \) has a local maximum at \( x = -6 \).
- \( f \) has a local minimum at \( x = 6 \).
- There are no critical points that are neither local minima nor maxima.

**(f) Concavity:**
- The function is not concave up for any \( x \).
- The function is not concave down for any \( x \).

**(g) Inflection Points:**
- There are no inflection points.

This analysis provides a comprehensive understanding of the behavior of the given function, including its asymptotic behavior, symmetry, and critical points.

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