Find the relative extrema of the function, if they exist.
\[
f(x)=-3 x^{3}+9
\]
A. Relative minimum at $(0,9)$
B. Relative maximum at $(0,-3)$
C. Relative maximum at $(0,9)$
D. No relative extrema exist.

Step 1 ：Find the derivative of the function \(f(x) = -3x^3 + 9\), which is \(f'(x) = -9x^2\).

Step 2 ：Set the derivative equal to zero and solve for x to find the critical points: \(-9x^2 = 0\) gives the critical point \(x = 0\).

Step 3 ：Substitute the critical point back into the original function to find the corresponding y-value: \(f(0) = -3(0)^3 + 9 = 9\). So, the critical point is at \((0,9)\).

Step 4 ：Find the second derivative of the function, which is \(f''(x) = -18x\).

Step 5 ：Evaluate the second derivative at the critical point: \(f''(0) = -18(0) = 0\). The second derivative test is inconclusive.

Step 6 ：However, since the function is a cubic function and the coefficient of the highest degree term is negative, we know that the function decreases as x approaches positive infinity and increases as x approaches negative infinity. Therefore, the critical point at x=0 is a relative maximum.

Step 7 ：Final Answer: The relative extrema of the function is a relative maximum at \((0,9)\). So, the correct answer is \(\boxed{\text{(C) Relative maximum at }(0,9)}\).