To identify each of the functions $f$, $f'$, and $f''$ from the graphs, let’s analyze the relationships between these functions and their graphical behaviors:

1. Function $f$:

   • $f$ is the original function, which should have relative maxima or minima where its derivative $f'$ crosses the x-axis (where $f' = 0$).
   • The concavity of $f$ (whether it curves upwards or downwards) is determined by $f''$, the second derivative.

2. First Derivative $f'$:

   • The first derivative $f'$ gives the slope of the tangent to the curve $f$. Where $f'$ crosses the x-axis, $f$ has a maximum or minimum (i.e., critical points).
   • $f'$ will be positive where $f$ is increasing and negative where $f$ is decreasing.

3. Second Derivative $f''$:

   • The second derivative $f''$ tells us about the concavity of $f$. Where $f''$ is positive, $f$ is concave up, and where $f''$ is negative, $f$ is concave down.
   • Points where $f''$ crosses the x-axis indicate inflection points of $f$, where the concavity changes.

Now, based on the graph:

• Graph $a$ (red):

  • This graph looks smooth and shows large changes in its slope. It appears to have a local maximum and minimum, which corresponds to critical points where $f'$ (the derivative) should be zero.
  • Hence, $a$ is likely $f$, the original function.

• Graph $b$ (green):

  • This graph crosses the x-axis at the points where $a$ (red) has maxima and minima, which suggests that $b$ represents the derivative $f'$.
  • $b$ is likely $f'$, the first derivative.

• Graph $c$ (blue):

  • This graph crosses the x-axis at the inflection points of $a$ (red), where the concavity of $a$ changes. This indicates that $c$ represents $f''$, the second derivative.
  • $c$ is likely $f''$, the second derivative.

Final Identification:

• $a$ is $f$
• $b$ is $f'$
• $c$ is $f''$

Would you like further clarification on any of these points or a deeper explanation? Here are some questions to extend your understanding:

1. How do we find the critical points of a function using its derivative?
2. What role does the second derivative play in determining the concavity of a function?
3. How can inflection points be found using the second derivative?
4. How does the first derivative test work to classify maxima and minima?
5. Can you explain the connection between the second derivative and acceleration in physics?

Tip: When analyzing graphs of $f$, $f'$, and $f''$, always check where the first derivative equals zero to find the extrema of $f$, and where the second derivative changes sign to find inflection points.