A different way to reach the answer is to test each option against the known roles of these two operators in finite-difference calculus.
The option stating central difference gives the second derivative while averaging gives the first derivative is false because $\delta f/h$ is the standard first-derivative estimator, not a second-derivative one; a second-derivative estimator instead uses $\delta^2 f(x)/h^2 = [f(x+h)-2f(x)+f(x-h)]/h^2$.
The option claiming both operators approximate the derivative with the same order of accuracy is false because the averaging operator $\mu$ does not approximate any derivative in the first place, it approximates $f(x)$ itself with an $O(h^2)$ curvature term, so comparing an order of accuracy for the derivative does not even apply to it.
The option stating averaging gives the second derivative while central difference gives the first derivative is also false for the same reason, since $\mu f(x)$ tends to $f(x)$, not to any derivative.
By elimination, only the statement that the central difference operator is more accurate than the averaging operator for approximating the derivative is consistent with the definitions, since $\mu$ is not a derivative operator at all while $\delta/h$ is an $O(h^2)$-accurate derivative operator.
\[\boxed{\text{The central difference operator is more accurate than the averaging operator for approximating the derivative.}}\]