The problem asks which graph represents the derivative $f'(x)$ of the function $f(x)$ shown in the top-right corner (the blue curve).

Analysis of $f(x)$ (blue curve):

1. The graph of $f(x)$ appears to have a smooth peak at $x = 0$, where the slope of the curve decreases as $x$ approaches 0 and then increases. The curve is symmetric around the y-axis, resembling a bell shape.
2. This suggests that:
   • The derivative $f'(x)$ will be 0 at $x = 0$ because the tangent line to the curve at the peak (maximum) is horizontal.
   • The slope is positive for $x < 0$ and negative for $x > 0$, indicating a change in sign of the derivative.

Steps to Identify the Derivative:

• $f'(x) = 0$ at $x = 0$ (the peak of the curve).
• As $x$ moves away from 0 (both left and right), the slope of the curve decreases (positive on the left, negative on the right).
• This means that $f'(x)$ should start positive for $x < 0$, decrease to 0 at $x = 0$, and become negative for $x > 0$.

Inspection of Options:

• Option (B) seems like the best candidate for $f'(x)$ because:
  • It crosses the x-axis at $x = 0$, indicating $f'(0) = 0$.
  • It has positive values for $x < 0$ and negative values for $x > 0$, matching the behavior of the slope of $f(x)$.

Thus, the correct answer is (B).

Would you like further details, or have any questions on this?

Here are 5 related questions to explore:

1. How do you determine the behavior of a derivative graph from the original function?
2. What is the significance of the derivative being zero at a point?
3. How does symmetry in a function affect its derivative?
4. Why is the derivative of a maximum or minimum point always zero?
5. What types of functions have derivative graphs that cross the x-axis?

Tip: When analyzing graphs, always check where the slope (tangent) is zero, as this gives crucial points for understanding the derivative.