The problem is asking to match the graphs of functions (labeled A through D) with the graphs of their respective derivatives (labeled 1 through 4).

Here's how to approach the problem:

Key observations when matching graphs:

1. Critical points of the original function:

   • At points where the original function (A–D) has a local maximum or minimum, the derivative should cross the x-axis (because the slope is zero at those points).

2. Slope of the original function:

   • When the original function is increasing, the derivative is positive (above the x-axis).
   • When the original function is decreasing, the derivative is negative (below the x-axis).

3. Shape of the derivative:

   • If the original function is linear (a straight line), the derivative will be a constant (a horizontal line).
   • If the original function is quadratic, the derivative will be linear.

Now, let’s match them based on these rules:

Step-by-step matching:

1. Graph A:

   • This is a cubic-like function with a local minimum and a local maximum, which means its derivative will have two roots (crosses the x-axis twice).
   • Matches Derivative 2: This derivative has two x-intercepts, corresponding to the two critical points in graph A.

2. Graph B:

   • This is a linear function, meaning its derivative should be a constant value.
   • Matches Derivative 4: The derivative of graph B is a constant, which corresponds to the horizontal line in graph 4.

3. Graph C:

   • This function has a point of inflection where the slope flattens and then changes. It looks cubic in nature, but smoother.
   • Matches Derivative 3: The graph has a more complex behavior near the origin, showing rapid changes that match the behavior of graph C.

4. Graph D:

   • This graph has a pattern of oscillations with three critical points (two maxima and one minimum).
   • Matches Derivative 1: The derivative graph has three x-intercepts, which corresponds to the critical points in graph D.

Final matches:

1. A → 2
2. B → 4
3. C → 3
4. D → 1

Would you like further explanations on how the derivatives were identified, or do you have any other questions?

Relative Questions:

1. What is the general relationship between the function and its derivative regarding increasing and decreasing behavior?
2. How does the second derivative of a function relate to the concavity of the original function?
3. How can critical points help in identifying maxima, minima, or inflection points of a function?
4. Why does a constant function have a derivative of zero, and what does this signify in terms of the slope?
5. How can you determine the number of turning points in a function based on its derivative?

Tip:

Always examine the behavior of the slope at the critical points (maxima, minima, or inflection points) to understand how the graph of the derivative behaves.