The problem asks to identify which of the given graphs represents the derivative of the function $f(x)$, whose graph is shown at the top.

Analysis:

1. The graph of $f(x)$ appears to have a peak at $x = 0$, and its slope decreases on either side of the peak. This implies that the derivative $f'(x)$ should be zero at $x = 0$ (since the slope of $f(x)$ is zero at this point).
2. The slope of $f(x)$ is positive to the left of $x = 0$, becomes zero at $x = 0$, and turns negative to the right of $x = 0$. This means $f'(x)$ should transition from positive to negative as we pass through $x = 0$, which indicates that $f'(x)$ should cross the x-axis at $x = 0$.

Identifying the Correct Graph:

• Graph (A): This graph has a derivative that changes from positive to negative, but it doesn't cross the x-axis properly in a way that matches the given $f(x)$.
• Graph (B): This derivative graph behaves similarly to what we expect from $f'(x)$, with a positive slope to the left of $x = 0$, zero at $x = 0$, and a negative slope to the right of $x = 0$. This looks like the correct choice.
• Other graphs (C), (D), (E), (F), and (G) either show discontinuities, vertical asymptotes, or incorrect behavior at $x = 0$, which does not match the original function.

Conclusion:

The correct answer is (B), as it reflects the expected behavior of the derivative of the given graph $f(x)$.

Would you like a more detailed explanation of any specific part?

Related Questions:

1. What are the key properties of a derivative graph when analyzing critical points?
2. How does the behavior of a function around maxima and minima reflect in its derivative?
3. How do discontinuities in the original function affect the derivative graph?
4. What are the differences between graphical features of a derivative and the original function?
5. Why is the derivative zero at the peak of a graph, and how does this affect graph interpretation?

Tip:

When analyzing graphs of functions and their derivatives, always look for points where the slope changes dramatically, such as peaks, troughs, and inflection points. These indicate key transitions in the derivative.