https://file.aiquickdraw.com/mathbot/file/17272322217206kpzxied.jpg

To solve this problem, we need to analyze the function \( f(x) \), its first derivative \( f'(x) \), and its second derivative \( f''(x) \) at each of the labeled points A, B, C, D, and E. Here's the step-by-step analysis:

### 1. **Identifying Points of Interest**:
   - **A**: Point where \( f(x) \) is at a local minimum.
   - **B**: Point where \( f(x) \) crosses the x-axis (root of the function).
   - **C**: Point where \( f(x) \) is at a local maximum.
   - **D**: Another point where \( f(x) \) crosses the x-axis.
   - **E**: Point on the downward slope after point \( D \).

### 2. **Determining Signs**:
   - **\( f(x) \)**:
     - Positive above the x-axis, Negative below the x-axis, Zero on the x-axis.
   - **\( f'(x) \)**:
     - Positive when the function is increasing, Negative when decreasing, Zero at local maxima or minima.
   - **\( f''(x) \)**:
     - Positive for concave up (cup-shaped), Negative for concave down (cap-shaped).

### 3. **Analysis of Points**:
   - **Point A**: 
     - \( f(x) \) is Negative (below x-axis).
     - \( f'(x) \) is Zero (local minimum).
     - \( f''(x) \) is Positive (concave up).
   - **Point B**: 
     - \( f(x) \) is Zero (crosses x-axis).
     - \( f'(x) \) is Positive (increasing slope).
     - \( f''(x) \) is Positive (concave up).
   - **Point C**:
     - \( f(x) \) is Positive (above x-axis).
     - \( f'(x) \) is Zero (local maximum).
     - \( f''(x) \) is Negative (concave down).
   - **Point D**:
     - \( f(x) \) is Zero (crosses x-axis).
     - \( f'(x) \) is Negative (decreasing slope).
     - \( f''(x) \) is Negative (concave down).
   - **Point E**:
     - \( f(x) \) is Negative (below x-axis).
     - \( f'(x) \) is Negative (decreasing slope).
     - \( f''(x) \) is Positive (concave up).

### 4. **Final Signs Table**:
| Point | \( f(x) \) | \( f'(x) \) | \( f''(x) \) |
|-------|-----------|-------------|-------------|
| A     | N         | 0           | P           |
| B     | 0         | P           | P           |
| C     | P         | 0           | N           |
| D     | 0         | N           | N           |
| E     | N         | N           | P           |

Let me know if you have any questions or need further details!

### Questions:
1. What determines whether \( f'(x) \) is positive or negative at a point?
2. Why is \( f'(x) \) zero at local maxima and minima?
3. How can you tell if \( f''(x) \) is positive or negative by looking at the graph?
4. What is the significance of the points where \( f(x) \) crosses the x-axis?
5. How do concavity and inflection points relate to the sign of \( f''(x) \)?

### Tip:
When analyzing graphs, always pay close attention to the curvature to determine the sign of the second derivative.

At exactly two of the labeled points in the figure below, which shows a function f, the derivative f' is 0; the second derivative f'' is not 0 at any of the labeled points. For each of f, f', and f'' at each marked point, enter the correct sign: 'P' for positive, 'N' for negative, or '0' for zero.

Solve a problem analyzing the signs of a function f, its first derivative f', and its second derivative f'' at various points on a curve. Learn how to determine whether each is positive, negative, or zero based on the graph. Suitable for calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation