Final answer:
The derivative of f(6) equaling zero means that the function has a horizontal tangent at x = 6, implying c. f(x) is continuous at x = 6. It does not provide information about the function's behavior at points other than x = 6.
Step-by-step explanation:
If the derivative of f(6) equals 0, it indicates that at x = 6, the function f(x) has a horizontal tangent line and therefore, at this point, the function is neither increasing nor decreasing. However, this fact alone does not provide information about the values of the function at other points, such as f(6.5) or f(5.5), nor does it imply that the differences in function values like f(7) - f(5) would be equal to 0.
The correct answer here is that c. f(x) is continuous at x = 6. Having a derivative at x = 6 means the function must be differentiable at that point, and a necessary condition for a function to be differentiable is that it must be continuous at that point. However, simply being differentiable at a point does not guarantee continuity at other points or specific function values.