### Explanation of Concavity and Inflection PointsTo determine the intervals on which the function is concave up or down and find the points of inflection, we will examine the given function:\[ y = 19x^2 + \ln(x) \quad (x > 0) \]**Instructions:**- Provide intervals in the form $(\ast, \ast)$. Use the symbol $\infty$ for infinity, $\cup$ for combining intervals, and an appropriate type of parenthesis "$( \, )$", "$[ \, ]$", "$[ \, )$", or "$( \, ]$", depending on whether the interval is open or closed. Enter $\emptyset$ if the interval is empty.- Provide points of inflection as a comma-separated list of $(x, y)$ ordered pairs. If the function does not have any inflection points, enter **DNE** (Does Not Exist).- Use exact values for all responses.**Graph and Diagram Explanation:**There are no graphs or diagrams provided in the image. The form consists of three input fields for users to fill out based on their analysis:1. **Concave Up Interval:** - Input the interval(s) where the function is concave up.2. **Concave Down Interval:** - Input the interval(s) where the function is concave down.3. **Points of Inflection:** - Input the (x, y) coordinates of the points of inflection.**User Input Fields:**1. **Concave Up:** \[ \text{Concave up:} \quad \_\_\_\_\_\_\_ \]2. **Concave Down:** \[ \text{Concave down:} \quad \_\_\_\_\_\_\_ \]3. **Points of Inflection:** \[ (x, y) = \quad \_\_\_\_\_\_\_ \]Users are expected to utilize their calculus knowledge to determine the concavity (second derivative test) and points of inflection (where the second derivative changes sign) of the provided function.

For the point of inflection, we solve y"=0

Determine the intervals on which the function is concave up or down and find the points of inflection. y = 19x + In(x) (x > 0) Provide intervals in the form (*, *). Use the symbol o for infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", or "]", depending on whether the interval is open or closed. Enter Ø if the interval is empty. Provide points of inflection as a comma-separated list of (x, y) ordered pairs. If the function does not have any inflection points, enter DNE. Use exact values for all responses.

Determine the intervals on which the function is concave up or down and find the points of inflection. y = 19x + In(x) (x > 0) Provide intervals in the form (, ). Use the symbol o for infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", or "]", depending on whether the interval is open or closed. Enter Ø if the interval is empty. Provide points of inflection as a comma-separated list of (x, y) ordered pairs. If the function does not have any inflection points, enter DNE. Use exact values for all responses.

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

### Explanation of Concavity and Inflection Points

To determine the intervals on which the function is concave up or down and find the points of inflection, we will examine the given function:

\[ y = 19x^2 + \ln(x) \quad (x > 0) \]

**Instructions:**
- Provide intervals in the form $(\ast, \ast)$. Use the symbol $\infty$ for infinity, $\cup$ for combining intervals, and an appropriate type of parenthesis "$( \, )$", "$[ \, ]$", "$[ \, )$", or "$( \, ]$", depending on whether the interval is open or closed. Enter $\emptyset$ if the interval is empty.
- Provide points of inflection as a comma-separated list of $(x, y)$ ordered pairs. If the function does not have any inflection points, enter **DNE** (Does Not Exist).
- Use exact values for all responses.

**Graph and Diagram Explanation:**
There are no graphs or diagrams provided in the image. The form consists of three input fields for users to fill out based on their analysis:

1. **Concave Up Interval:**
- Input the interval(s) where the function is concave up.

2. **Concave Down Interval:**
- Input the interval(s) where the function is concave down.

3. **Points of Inflection:**
- Input the (x, y) coordinates of the points of inflection.

**User Input Fields:**

1. **Concave Up:**
\[ \text{Concave up:} \quad \_\_\_\_\_\_\_ \]

2. **Concave Down:**
\[ \text{Concave down:} \quad \_\_\_\_\_\_\_ \]

3. **Points of Inflection:**
\[ (x, y) = \quad \_\_\_\_\_\_\_ \]

Users are expected to utilize their calculus knowledge to determine the concavity (second derivative test) and points of inflection (where the second derivative changes sign) of the provided function.

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