To analyze this question, let’s examine the graph of $f(x)$:

The graph shows:

1. A peak around $x \approx 0.5$, indicating a local maximum where the derivative $f'(x) = 0$.
2. A valley around $x \approx 2$, indicating a local minimum where $f'(x) = 0$.
3. Another point where the curve flattens around $x \approx 3.5$, but it’s neither a maximum nor a minimum, suggesting it’s an inflection point where $f'(x) = 0$.

Option Analysis

1. Option 1: "The derivative is zero at two values of $x$, one is a local maximum while the other is a local minimum."

   • This is correct because at $x \approx 0.5$, there's a local maximum, and at $x \approx 2$, there's a local minimum.

2. Option 2: "The derivative is zero only at one value of $x$ where it is a local minimum."

   • This is incorrect as the derivative is zero at more than one point.

3. Option 3: "The derivative is zero at two values of $x$, one is a local maximum on the interval while the other is neither a local maximum nor a minimum."

   • This is incorrect since one of the points where the derivative is zero is indeed a local minimum.

4. Option 4: "The derivative is zero at two values of $x$, one is a local minimum on the interval while the other is neither a local maximum nor a minimum."

   • This option is incorrect because one point is a local maximum.

5. Option 5: "The derivative is zero at two values of $x$, both being local maxima."

   • This is incorrect as one is a local maximum and the other is a local minimum.

Conclusion

The correct answer is Option 1: "The derivative is zero at two values of $x$, one is a local maximum while the other is a local minimum."

Would you like further details on this? Here are some related questions that might help deepen your understanding:

1. What defines a critical point, and how does it differ from an inflection point?
2. How do we determine whether a point is a local maximum, minimum, or neither?
3. What are the implications of a zero derivative in a function's graph?
4. How can we identify inflection points in a function?
5. Why might the second derivative test be useful for confirming maxima and minima?

Tip: To confirm whether a point is a maximum, minimum, or inflection point, consider using both the first and second derivative tests.