The question asks to determine which of the graphs corresponds to $f$, $f'$, and $f''$, based on the given graphs.

Here's how we can identify each graph:

1. Graph of $f$ (the original function):

   • The function $f$ should have the general shape of a continuous smooth curve, likely crossing the x-axis at some points, and should exhibit turning points (local maxima and minima) where its derivative $f'$ equals zero.
   • The solid green curve appears to represent $f$. It has a turning point around $x = 0$ and another one around $x = 2$, typical of a cubic or similar function.

2. Graph of $f'$ (the first derivative):

   • The graph of $f'$ represents the slope of $f$. It should be zero at the points where $f$ has local maxima or minima (turning points). It also changes sign at these points: positive slope where $f$ is increasing, negative slope where $f$ is decreasing.
   • The dotted brown curve fits this description. It crosses the x-axis at the turning points of $f$, indicating where the slope of $f$ is zero. It also matches the increasing and decreasing behavior of $f$.

3. Graph of $f''$ (the second derivative):

   • The graph of $f''$ represents the concavity of $f$. It indicates how the slope of $f$ is changing. $f''$ is positive where $f$ is concave up and negative where $f$ is concave down. The second derivative is zero at inflection points.
   • The dashed blue curve seems to correspond to $f''$, as it shows where the concavity of $f$ changes. It crosses the x-axis where $f$ changes concavity.

Conclusion:

• Green solid curve: $f$ (the original function)
• Brown dotted curve: $f'$ (the first derivative)
• Blue dashed curve: $f''$ (the second derivative)

Would you like further clarification on these concepts or the steps involved?

Related Questions:

1. How can you determine concavity and inflection points from the graph of $f$?
2. What is the significance of $f'(x) = 0$ in the context of turning points?
3. How does the behavior of $f'$ relate to the increasing and decreasing intervals of $f$?
4. How do we interpret the sign changes in $f''(x)$ in terms of the graph of $f$?
5. What is the relationship between the steepness of $f$ and the value of $f'(x)$?

Tip:

Remember that the graph of $f'$ is crucial for understanding the behavior of $f$, especially in identifying maxima, minima, and intervals of increase or decrease!