In this problem, you are given three functions (A, B, and C) and asked to match them with the appropriate graphs of their derivatives, $f'(x)$, from the choices I to VIII.

Here’s how you can approach the problem:

Step-by-step approach:

1. Look at the critical points of $f(x)$:

   • The critical points (where $f'(x) = 0$) occur at the peaks, troughs, or flat points of the graph of $f(x)$. These will correspond to the points where the graph of $f'(x)$ crosses the x-axis (i.e., where $f'(x) = 0$).

2. Analyze the behavior around critical points:

   • If $f(x)$ is increasing, $f'(x) > 0$.
   • If $f(x)$ is decreasing, $f'(x) < 0$.

3. Match this with the behavior in the derivative graphs:

   • For each function, determine where $f(x)$ is increasing, decreasing, or has a critical point, and match this behavior with the corresponding graph.

Analysis of Functions:

Function A:

• Shape: Function A starts decreasing, then increases after a minimum point.

• Derivative behavior:

  • Decreasing means $f'(x) < 0$ on the left.
  • After the minimum point, $f'(x) > 0$.
  • There is a critical point (where $f'(x) = 0$) at the minimum.

  Matching Derivative: This behavior matches Graph V, which shows $f'(x)$ going from negative to positive with a zero at a critical point.

Function B:

• Shape: Function B is a straight line, constantly decreasing.

• Derivative behavior:

  • Constantly decreasing means $f'(x)$ is a constant negative value.

  Matching Derivative: This matches Graph VII, which shows a constant negative value for $f'(x)$.

Function C:

• Shape: Function C is increasing, reaching a maximum, and then decreasing.

• Derivative behavior:

  • Increasing means $f'(x) > 0$.
  • Decreasing means $f'(x) < 0$.
  • There is a critical point (where $f'(x) = 0$) at the maximum.

  Matching Derivative: This behavior matches Graph VI, which shows $f'(x)$ going from positive to negative with a zero at the critical point.

Conclusion:

• Function A: V
• Function B: VII
• Function C: VI

Would you like further clarifications or explanations on any of these steps?

Here are some additional related questions:

1. How do you determine where a function has a critical point based on its graph?
2. How does the second derivative affect the concavity of a function?
3. What is the significance of the inflection points in relation to the graph of a function and its derivative?
4. Can a function have a horizontal tangent without having a critical point?
5. How do we interpret the behavior of the derivative for linear functions?

Tip: Always remember that the derivative's graph gives you the slope of the original function at every point. If you master reading slope changes, you'll master derivatives!