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This image shows graphs of several functions labeled \( y = f(x) \) and \( y = g(x) \). The task is to match each function \( y = f(x) \) with its corresponding derivative \( y = g(x) \) graph.

To match these correctly, let's analyze the features of each graph, specifically focusing on:

1. **The slope behavior of \( f(x) \)**: The derivative \( g(x) \) will have zeroes where \( f(x) \) has local maxima, minima, or points of inflection, as these represent points where the slope of \( f(x) \) changes.
2. **Positive and negative slopes**: In regions where \( f(x) \) is increasing, \( g(x) \) will be positive, and where \( f(x) \) is decreasing, \( g(x) \) will be negative.

Let's proceed by matching each \( y = f(x) \) graph (top row) to its corresponding \( y = g(x) \) graph (bottom row). Here is the step-by-step approach:

1. **First \( f(x) \) graph (far left):**
   - It has a minimum around \( x = -2 \) and a maximum near \( x = 2 \).
   - The derivative \( g(x) \) should pass through zero around \( x = -2 \) and \( x = 2 \).
   - The best match for this graph appears to be the second \( g(x) \) graph in the bottom row.

2. **Second \( f(x) \) graph (second from left):**
   - It has a minimum near \( x = -3 \) and a maximum near \( x = 1 \).
   - The derivative \( g(x) \) should have zeroes at these points.
   - The best match seems to be the fourth \( g(x) \) graph in the bottom row.

3. **Third \( f(x) \) graph (third from left):**
   - Shows a parabola opening downwards with a maximum point around \( x = 0 \).
   - The derivative \( g(x) \) should be a linear graph crossing the x-axis at \( x = 0 \).
   - The best match is the first \( g(x) \) graph in the bottom row.

4. **Fourth \( f(x) \) graph (far right):**
   - This function has a maximum near \( x = -1 \) and a minimum near \( x = 3 \).
   - The derivative \( g(x) \) should cross the x-axis at these points.
   - The best match is the third \( g(x) \) graph in the bottom row.

By matching based on these observations, here is the solution:

- **First \( f(x) \)** goes with **Second \( g(x) \)**
- **Second \( f(x) \)** goes with **Fourth \( g(x) \)**
- **Third \( f(x) \)** goes with **First \( g(x) \)**
- **Fourth \( f(x) \)** goes with **Third \( g(x) \)**

Would you like further clarification on any of these steps, or an explanation of the derivative properties? 

### Related Questions
1. How can we identify points of inflection in a function's graph?
2. What does the derivative graph tell us about the original function's concavity?
3. How do the critical points of \( f(x) \) correspond to the zeroes of \( g(x) \)?
4. How can we tell if a function is concave up or concave down based on its derivative?
5. What role do asymptotes play in matching functions with their derivatives?

### Tip
Always check where the slope of \( f(x) \) is zero to find potential matches for \( g(x) \), as these are key indicators of derivative behavior.

Nubrėžti funkcijų y = f(x) ir jų išvestinių y = g(x) grafikų poras pagal pateiktus grafikus.

Learn how to match graphs of functions y = f(x) with their derivative graphs y = g(x). This problem covers key calculus concepts such as derivatives, critical points, and slope analysis, suitable for students in Grades 10-12.

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