The derivative of a function at a particular point represents the slope of the tangent line to the curve at that point. When the derivative is zero, it means that the slope of the tangent line is horizontal, i.e., the tangent line is flat.

This typically happens at critical points of the function, which can be either:

1. Local maxima: Points where the function reaches a peak (highest point locally).
2. Local minima: Points where the function reaches a trough (lowest point locally).
3. Points of inflection (though at points of inflection, the derivative might not necessarily be zero).

In the context of the graph you provided:

• At point $a$: The curve reaches a local maximum. The slope changes from positive to negative, so the tangent at the peak is horizontal, making $f'(a) = 0$.
• At point $c$: The curve reaches a local minimum. The slope changes from negative to positive, so the tangent at the trough is also horizontal, making $f'(c) = 0$.

These horizontal tangents indicate that the rate of change of the function (the slope) at those points is zero, which is why the derivative is zero at these points.

Would you like more details or have any other questions?

Here are some additional related questions:

1. How can you determine if a critical point is a maximum, minimum, or a point of inflection?
2. What does the first derivative test tell you about the behavior of a function?
3. How does the second derivative help in understanding the curvature of a graph?
4. Can there be more than one point where the derivative is zero?
5. How would you interpret the derivative at a point where the curve is flat for an extended interval?

Tip: Remember, when the derivative is zero, it's important to check the surrounding points to determine if it's a maximum, minimum, or a point of inflection.