the second picture is part of the first picture, i could not fit it all in one screenshot. ### Graph of the Derivative of a Function#### Description:The figure represents the graph of the derivative, $ f' $, of a function $ f $ on the interval $[-3, 3]$.#### Graph Explanation:The graph features a sinusoidal curve that crosses the x-axis at three points: approximately $ x = -2 $, $ x = 0 $, and $ x = 2 $. The curve reaches its maximum point at $ x = -1 $ and its minimum point at $ x = 1 $.#### Determining Intervals of Increase and Decrease:To determine where $ f $ is increasing or decreasing, we observe where $ f' $, the derivative, is positive or negative, respectively. - $ f $ is increasing when $ f' $ is above the x-axis (positive).- $ f $ is decreasing when $ f' $ is below the x-axis (negative).#### Tasks:1. Determine the intervals on which $ f $ is increasing.2. Determine the intervals on which $ f $ is decreasing.(Use the symbol $ \cup $ for combining intervals, and the appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed.)---**Find the intervals where $ f $ is increasing:**\[ f \text{ is increasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]**Find the intervals where $ f $ is decreasing:**\[ f \text{ is decreasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]---### Solution Explanation:Use the graph to identify the intervals where the derivative is positive (above the x-axis) and negative (below the x-axis). From the graph:- $ f $ is increasing on intervals where $ f' $ > 0: - $ (-2, 0) \cup (2, 3) $ - $ f $ is decreasing on intervals where $ f' $ < 0: - $ (-3, -2) \cup (0, 2) $Therefore,\[ f \text{ is increasing on} \quad (-2, 0) \cup (2, 3) \]\[ f \text{ is decreasing on} \ ---### Determining Concavity Intervals#### Determine the intervals on which $ f $ is concave down.(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “$ ( $”, “\[ \]” depending on whether the interval is open or closed.)\[ f \text{ is concave down on } \underline{\hspace{400px}} \]---#### Determine the intervals on which $ f $ is concave up.(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “$ ( $”, “\[ \]” depending on whether the interval is open or closed.)\[ f \text{ is concave up on } \underline{\hspace{400px}} \]---### Explanation:Concavity refers to the direction of the curvature of a graph:- **Concave Down**: The graph of $ f $ is concave down where it curves downwards like an upside-down cup. This occurs where the second derivative $ f''(x) $ is less than zero ($ f''(x) < 0 $).- **Concave Up**: The graph of $ f $ is concave up where it curves upwards like a right-side-up cup. This occurs where the second derivative $ f''(x) $ is greater than zero ($ f''(x) > 0 $).When determining concavity intervals, use:- **Parentheses $ (\ ) $**: To denote open intervals (where the endpoint is not included).- **Brackets $[ ]$**: To denote closed intervals (where the endpoint is included).- **Union $ \cup $**: To combine multiple intervals.---This page is designed to help students understand and practice how to find and express the intervals where a function $ f $ is concave up or concave down.For additional assistance, review resources on calculus and concavity concepts.---

Answered: Image /qna-images/answer/c4941b04-05bc-4f5a-b0b4-d28ce1a9e7b0.jpg

Answered: The figure is the graph of the…

The figure is the graph of the derivative, f', of a function f on [-3,3]. 2+ 2 -2+ Determine the intervals on which ƒ is increasing.

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

the second picture is part of the first picture, i could not fit it all in one screenshot.

### Graph of the Derivative of a Function

#### Description:
The figure represents the graph of the derivative, $ f' $, of a function $ f $ on the interval $[-3, 3]$.

#### Graph Explanation:
The graph features a sinusoidal curve that crosses the x-axis at three points: approximately $ x = -2 $, $ x = 0 $, and $ x = 2 $. The curve reaches its maximum point at $ x = -1 $ and its minimum point at $ x = 1 $.

#### Determining Intervals of Increase and Decrease:
To determine where $ f $ is increasing or decreasing, we observe where $ f' $, the derivative, is positive or negative, respectively.

- $ f $ is increasing when $ f' $ is above the x-axis (positive).
- $ f $ is decreasing when $ f' $ is below the x-axis (negative).

#### Tasks:
1. Determine the intervals on which $ f $ is increasing.
2. Determine the intervals on which $ f $ is decreasing.

(Use the symbol $ \cup $ for combining intervals, and the appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed.)

---

**Find the intervals where $ f $ is increasing:**
\[ f \text{ is increasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]

**Find the intervals where $ f $ is decreasing:**
\[ f \text{ is decreasing on} \quad \_\_\_\_\_\_\_\_\_\_\_\_\_\ \]

---

### Solution Explanation:
Use the graph to identify the intervals where the derivative is positive (above the x-axis) and negative (below the x-axis). From the graph:

- $ f $ is increasing on intervals where $ f' $ > 0:
- $ (-2, 0) \cup (2, 3) $

- $ f $ is decreasing on intervals where $ f' $ < 0:
- $ (-3, -2) \cup (0, 2) $

Therefore,
\[ f \text{ is increasing on} \quad (-2, 0) \cup (2, 3) \]
\[ f \text{ is decreasing on} \

---

### Determining Concavity Intervals

#### Determine the intervals on which $ f $ is concave down.

(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “$ ( $”, “\[ \]” depending on whether the interval is open or closed.)

\[ f \text{ is concave down on } \underline{\hspace{400px}} \]

---

#### Determine the intervals on which $ f $ is concave up.

(Use the symbol ∪ for combining intervals, and an appropriate type of parenthesis “$ ( $”, “\[ \]” depending on whether the interval is open or closed.)

\[ f \text{ is concave up on } \underline{\hspace{400px}} \]

---

### Explanation:

Concavity refers to the direction of the curvature of a graph:

- **Concave Down**: The graph of $ f $ is concave down where it curves downwards like an upside-down cup. This occurs where the second derivative $ f''(x) $ is less than zero ($ f''(x) < 0 $).
- **Concave Up**: The graph of $ f $ is concave up where it curves upwards like a right-side-up cup. This occurs where the second derivative $ f''(x) $ is greater than zero ($ f''(x) > 0 $).

When determining concavity intervals, use:
- **Parentheses $ (\ ) $**: To denote open intervals (where the endpoint is not included).
- **Brackets $[ ]$**: To denote closed intervals (where the endpoint is included).
- **Union $ \cup $**: To combine multiple intervals.

---

This page is designed to help students understand and practice how to find and express the intervals where a function $ f $ is concave up or concave down.

For additional assistance, review resources on calculus and concavity concepts.

---

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