Given the graph of f "(X) below, which are possible graphs of f(x)? 10 2 2 2 -6 -8 10

Calculus: Early Transcendentals
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Chapter1: Functions And Models
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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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**Given the graph of f ''(x), which are possible graphs of f(x)?**

### Graph Descriptions:

1. **First Row - Graph of f ''(x):**
   - The graph is a straight diagonal line with a positive slope that passes through the origin (0,0) and extends through the points (3,3) on the right and (-3,-3) on the left. This suggests a constant positive rate of change for f '(x), implying f(x) is a quadratic function.

2. **Second Row - Graphs of Possible f(x):**

   - **Graph A (Left):**
     - A cubic-shaped curve. It starts below the x-axis, moves up passing through the x-axis around x = -1, peaks near x = 0, then dips slightly before rising sharply.

   - **Graph B (Right):**
     - A sinusoidal-like curve shifted upwards. It starts slightly below the x-axis, dips near x = -1, rises and flattens towards x = 3.

3. **Third Row - Graphs of Possible f(x):**

   - **Graph C (Left):**
     - A cubic-shaped curve. It begins above the x-axis, descends crossing the x-axis around x = -1, reaches a minimum, and then ascends sharply.

   - **Graph D (Right):**
     - An S-shaped curve extending from the upper left to the lower right. It crosses the x-axis at around x = 0 and continues downward.

### Analysis:
- Since f ''(x) is positive, indicating concavity is upward, the possible f(x) graphs should be consistent with having sections of increasing slope, suggesting the correct candidate will exhibit a predominantly increasing behavior or show convexity on interval sections.
Transcribed Image Text:**Given the graph of f ''(x), which are possible graphs of f(x)?** ### Graph Descriptions: 1. **First Row - Graph of f ''(x):** - The graph is a straight diagonal line with a positive slope that passes through the origin (0,0) and extends through the points (3,3) on the right and (-3,-3) on the left. This suggests a constant positive rate of change for f '(x), implying f(x) is a quadratic function. 2. **Second Row - Graphs of Possible f(x):** - **Graph A (Left):** - A cubic-shaped curve. It starts below the x-axis, moves up passing through the x-axis around x = -1, peaks near x = 0, then dips slightly before rising sharply. - **Graph B (Right):** - A sinusoidal-like curve shifted upwards. It starts slightly below the x-axis, dips near x = -1, rises and flattens towards x = 3. 3. **Third Row - Graphs of Possible f(x):** - **Graph C (Left):** - A cubic-shaped curve. It begins above the x-axis, descends crossing the x-axis around x = -1, reaches a minimum, and then ascends sharply. - **Graph D (Right):** - An S-shaped curve extending from the upper left to the lower right. It crosses the x-axis at around x = 0 and continues downward. ### Analysis: - Since f ''(x) is positive, indicating concavity is upward, the possible f(x) graphs should be consistent with having sections of increasing slope, suggesting the correct candidate will exhibit a predominantly increasing behavior or show convexity on interval sections.
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