https://file.aiquickdraw.com/mathbot/file/1728760157340uxjjpxsh.jpg

Let's walk through the question systematically: the task is to **match the functions (A)–(D) with their derivatives (I)–(III)**. 

### Step-by-Step Matching Process:
We’ll analyze the graphs of the functions and their behavior to identify the derivatives.

1. **Graph (A)**: 
   - **Shape:** A downward-opening parabola.
   - **Behavior:** It has a maximum at \(x = 0\), meaning the slope decreases on either side of the peak.
   - **Derivative:** A downward-sloping linear graph.
   - **Match:** This corresponds to **Graph (III)** (negative linear slope).

2. **Graph (B):**
   - **Shape:** A straight line with a positive slope.
   - **Behavior:** The slope is constant for all \(x\).
   - **Derivative:** The derivative of a linear function with constant slope is a horizontal line (the slope itself).
   - **Match:** This corresponds to **Graph (I)** (horizontal line).

3. **Graph (C):**
   - **Shape:** A curve that increases, peaks, and then decreases.
   - **Behavior:** The slope changes from positive to negative, indicating the derivative must be zero at the peak and changing sign.
   - **Derivative:** A parabola (opens upward or downward) fits this behavior.
   - **Match:** This corresponds to **Graph (II)** (upward-opening parabola).

4. **Graph (D):**
   - **Shape:** A downward-opening parabola (like (A)).
   - **Behavior:** Same as Graph (A), it has a maximum at the peak.
   - **Derivative:** Since it’s also a parabola with the same behavior, the derivative will again correspond to **Graph (III)**.

### Final Matches:
- **(A) → (III)**  
- **(B) → (I)**  
- **(C) → (II)**  
- **(D) → (III)**  

### Explanation of Repeated Derivatives:
- Both **(A) and (D)** are downward-opening parabolas, so their slopes behave the same way. This results in **the same derivative for both (III)**.

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Do you have any further questions or need clarifications?

### 5 Follow-up Questions:
1. What is the derivative of a cubic function, and how would its graph look?
2. How do we interpret the concavity of a graph using second derivatives?
3. Can you explain why the derivative of a horizontal line is zero?
4. How can you find the point of inflection using derivatives?
5. What happens when a function's derivative is undefined at a point?

### Tip:
When matching graphs to derivatives, always look at the slopes (increasing/decreasing) and the peaks or troughs for zero slopes—they reveal where the derivative must be zero.

Match the functions in graphs (A)–(D) with their derivatives (I)–(III). Note that two of the functions have the same derivative. Explain why.

In this calculus problem, learn how to match functions with their corresponding derivatives based on graph analysis. Includes step-by-step reasoning for matching parabolas, linear functions, and more. Suitable for high school calculus students.

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