In the image, the problem asks to identify the functions $f(x)$, $f'(x)$, and $f''(x)$ from three polynomial curves. These functions are related by differentiation, with:

• $f(x)$ being the original function (orange),
• $f'(x)$ being the first derivative (blue), and
• $f''(x)$ being the second derivative (green).

Here's how to approach the problem:

1. Start with the derivative behaviors:

   • $f'(x)$ (blue) must be the slope of $f(x)$. Wherever $f(x)$ has a horizontal tangent (i.e., local maxima or minima), $f'(x)$ should cross the x-axis (since the slope is zero).
   • Similarly, the graph of $f''(x)$ (green) must be positive where $f'(x)$ is increasing and negative where $f'(x)$ is decreasing, and it crosses the x-axis where $f'(x)$ has horizontal tangents.

2. Analyze the curves:

   • Look at the black solid curve: It appears to have turning points (local maxima/minima), which are key to finding where the derivative crosses the x-axis. This likely corresponds to $f(x)$.
   • The blue dashed curve crosses the x-axis at points where the black solid curve has maxima or minima. Therefore, this blue dashed curve represents $f'(x)$, the first derivative.
   • The dotted black curve changes signs at the same points where the blue dashed curve has maxima or minima. Hence, this curve is likely $f''(x)$, the second derivative.

Conclusion:

• The black solid curve represents $f(x)$ (orange).
• The blue dashed curve represents $f'(x)$ (blue).
• The black dotted curve represents $f''(x)$ (green).

Let me know if you'd like more clarification or details about the solution.

Follow-up questions:

1. Why does $f'(x)$ cross the x-axis where $f(x)$ has local maxima or minima?
2. How do the signs of $f''(x)$ relate to the concavity of $f(x)$?
3. What happens to $f'(x)$ at inflection points of $f(x)$?
4. How can the behavior of $f'(x)$ help determine where $f(x)$ is increasing or decreasing?
5. How does $f''(x)$ help predict the shape of $f(x)$?

Tip:

To analyze the relationship between $f(x)$, $f'(x)$, and $f''(x)$, always start by identifying where the slopes of the curves are zero (for $f'(x)$) and where the slopes are changing the fastest (for $f''(x)$).