The image provided shows three different curves, each in a distinct color: red, blue, and green. We are tasked with identifying which curve corresponds to $f(x)$, $f'(x)$, and $g(x)$.

Here's how we can approach this:

1. Derivative Analysis:

   • The function $f'(x)$ is the derivative of $f(x)$, which means that $f'(x)$ should indicate the slope of $f(x)$ at each point.
   • When $f(x)$ has a maximum or minimum (where the slope is 0), $f'(x)$ should cross the x-axis.

2. Identifying the curves:

   • Blue curve: The blue curve appears to have a smooth shape with a peak and a trough, indicating a likely candidate for $f(x)$. This function is continuous, and at its peaks and troughs, the slope should be zero. So $f'(x)$ must cross the x-axis at these points.
   • Red curve: This curve crosses the x-axis at the points where the blue curve has peaks or troughs, indicating that it could be the derivative $f'(x)$. The red curve also changes sign (positive to negative and vice versa) where the blue curve has extreme values, which further supports this idea.
   • Green curve: The green curve looks different in structure and does not seem to correlate with the behavior of the blue curve's slope. This indicates that it is likely the third function $g(x)$, independent of $f(x)$ and $f'(x)$.

Conclusion:

• The blue curve represents $f(x)$,
• The red curve represents $f'(x)$,
• The green curve represents $g(x)$.

Would you like further details on how derivatives work, or any other aspect of this? Let me know if you have any questions.

Here are 5 related questions:

1. How can we determine the points of inflection of $f(x)$ using $f'(x)$?
2. What is the relationship between the concavity of $f(x)$ and the sign of $f'(x)$?
3. How do you graphically identify critical points (maxima, minima) on $f(x)$ using $f'(x)$?
4. Can we predict the behavior of $f(x)$ near its critical points just by looking at $f'(x)$?
5. What properties distinguish a function from its derivative graphically?

Tip: Always check where the derivative crosses the x-axis, as these points often correspond to critical points (peaks or troughs) in the original function.