Please give me the solution with an explanation.Thank you! ### Using Derivatives to Sketch the Graph of a FunctionIn this post, you will use the first and second derivatives of a function (along with a few other pieces of information) to sketch the graph of a function.#### Sketch the Graph of a Single Function \( f(x) \)The function \( f(x) \) should satisfy all of the following conditions. Use the techniques learned in this course to do so.**First Derivative:**\[ f'(x) = \frac{16(x + 2)}{(6 - x)^3} \]**Second Derivative:**\[ f''(x) = \frac{32(x + 6)}{(6 - x)^4} \]**Domain:**The domain of \( f \) is \( (-\infty, 6) \cup (6, \infty) \)**Vertical Asymptote:**There is a vertical asymptote at \( x = 6 \)**Limits:**\[ \lim_{{x \to -\infty}} f(x) = 1 \]\[ \lim_{{x \to \infty}} f(x) = 1 \]**Point:**\[ f(2) = 1 \]**Instructions:**1. **Sketch the Graph:** - Using the first and second derivatives, sketch the graph of the function. 2. **Label the Asymptotes:** - Identify and label the equation of the vertical asymptote \( x = 6 \). 3. **Identify Local Extrema:** - Determine and label the locations of any local maxima or minima. 4. **Determine Intervals:** - Explicitly state the intervals where the function is increasing and decreasing. - Identify the intervals of concavity. 5. **Inflection Points:** - Determine the x-coordinates of any inflection points and label them on the graph.By following these steps, you can effectively sketch the graph of the function \( f(x) \) using its first and second derivatives to gather the necessary information about its behavior and characteristics.

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f (x) Sketch the graph of a single function that satisfies all of the following conditions. Use the techniques that we have learned in this course to do SO. f (x)= 16(x + 2) (6 − x)³ 32(x + 6) f" (x) = (6 - x) 4 The domain of fis (-∞, 6) U (6,∞) X = lim f(x) = 1, lim f(x) = 1 x118 x→∞ ƒ(2) = 1 After you have sketched the graph, label the equations of the asymptotes, as well as the locations of any local extrema. Then, explicitly state the intervals of increase and decrease, the intervals of concavity, and the x-coordinates of any inflection points. 6 is a vertical asymptote of f

f (x) Sketch the graph of a single function that satisfies all of the following conditions. Use the techniques that we have learned in this course to do SO. f (x)= 16(x + 2) (6 − x)³ 32(x + 6) f" (x) = (6 - x) 4 The domain of fis (-∞, 6) U (6,∞) X = lim f(x) = 1, lim f(x) = 1 x118 x→∞ ƒ(2) = 1 After you have sketched the graph, label the equations of the asymptotes, as well as the locations of any local extrema. Then, explicitly state the intervals of increase and decrease, the intervals of concavity, and the x-coordinates of any inflection points. 6 is a vertical asymptote of f

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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Please give me the solution with an explanation. Thank you!

### Using Derivatives to Sketch the Graph of a Function

In this post, you will use the first and second derivatives of a function (along with a few other pieces of information) to sketch the graph of a function.

#### Sketch the Graph of a Single Function \( f(x) \)

The function \( f(x) \) should satisfy all of the following conditions. Use the techniques learned in this course to do so.

**First Derivative:**
\[ f'(x) = \frac{16(x + 2)}{(6 - x)^3} \]

**Second Derivative:**
\[ f''(x) = \frac{32(x + 6)}{(6 - x)^4} \]

**Domain:**
The domain of \( f \) is \( (-\infty, 6) \cup (6, \infty) \)

**Vertical Asymptote:**
There is a vertical asymptote at \( x = 6 \)

**Limits:**
\[ \lim_{{x \to -\infty}} f(x) = 1 \]
\[ \lim_{{x \to \infty}} f(x) = 1 \]

**Point:**
\[ f(2) = 1 \]

**Instructions:**

1. **Sketch the Graph:**
- Using the first and second derivatives, sketch the graph of the function.

2. **Label the Asymptotes:**
- Identify and label the equation of the vertical asymptote \( x = 6 \).

3. **Identify Local Extrema:**
- Determine and label the locations of any local maxima or minima.

4. **Determine Intervals:**
- Explicitly state the intervals where the function is increasing and decreasing.
- Identify the intervals of concavity.

5. **Inflection Points:**
- Determine the x-coordinates of any inflection points and label them on the graph.

By following these steps, you can effectively sketch the graph of the function \( f(x) \) using its first and second derivatives to gather the necessary information about its behavior and characteristics.

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