Final answer:
The error in the approximation F"(x) using the finite difference method is related to the truncation error in Taylor series, typically O(h^4), but due to a likely typo in the question, O(h^2) would be the best guess from the options provided.
Step-by-step explanation:
The error involved in the approximation F"(x) given by [f(x+4h)-2f(x)+f(x-4h)]/16h^2 corresponds to the truncation error of a finite difference approximation to the second derivative of a function. This error can be analyzed using Taylor series expansions of f(x+4h) and f(x-4h) around f(x). If the function f is sufficiently smooth and h is small, the leading term of the error will typically be proportional to a power of h. The correct answer would be the order of O(h^4), which indicates the error term includes fourth powers of h and higher. However, this is not one of the options provided. Among the available options, the closest correct approximation would be O(h^2) as it implies the error has a quadratic relationship with h. None of the provided erroneous options match the correct order of the truncation error for the approximation given, but O(h^2) would be an educated guess, assuming a typographical mistake in the question.