9 Let $f(x)=x^{3}-\left(k^{2}-4\right) x+1$ For which of the values of $k$ below is $f(x)$ many-to-one? (A) $k=2$ (B) $k=-2$ (C) $k=1$ (D) $k=-4$

Step 1 ：Let \(f(x) = x^3 - (k^2 - 4)x + 1\)

Step 2 ：Find the critical points of \(f(x)\) by taking the derivative and setting it to 0:

Step 3 ：\(f'(x) = -k^2 + 3x^2 + 4\)

Step 4 ：Critical points: \(x = \pm\frac{\sqrt{3k^2 - 12}}{3}\)

Step 5 ：Check if there are multiple critical points for any of the given values of k:

Step 6 ：\(k = 2, -2, 1, -4\)

Step 7 ：Multiple critical points for \(k = 1\) and \(k = -4\)

Step 8 ：\(\boxed{\text{Final Answer: (C) } k = 1 \text{ and (D) } k = -4}\)