Find \( (A) f^{\prime}(x) \), (B) the partition numbers for \( f^{\prime} \), and \( (C) \) the critical numbers of \( f \), \( f(x)=x^{3}-12 x-8 \) (A) \( f^{\prime}(x)=\square \)

\( \textbf{Step 1. Differentiate } f(x) \) We start with \[ f(x)=x^3-12x-8. \] Differentiate term-by-term: \[ f'(x)=\frac{d}{dx}(x^3)-\frac{d}{dx}(12x)-\frac{d}{dx}(8)=3x^2-12. \] Thus, \[ \boxed{f'(x)=3x^2-12}. \] \( \textbf{Step 2. Find the Partition Numbers for } f' \) To find the partition numbers, we set \( f'(x)=0 \): \[ 3x^2-12=0. \] Divide both sides by \( 3 \): \[ x^2-4=0. \] Factorize: \[ (x-2)(x+2)=0. \] Thus, the solutions are: \[ x-2=0 \quad \Rightarrow \quad x=2, \] \[ x+2=0 \quad \Rightarrow \quad x=-2. \] So, the partition numbers are \[ \boxed{x=-2 \text{ and } x=2}. \] \( \textbf{Step 3. Find the Critical Numbers of } f \) Critical numbers occur when \( f'(x)=0 \) or is undefined. Since \( f(x) \) is a polynomial, \( f'(x) \) is defined for all \( x \). Thus, the critical numbers are the solutions to \( f'(x)=0 \), which are: \[ \boxed{x=-2 \text{ and } x=2}. \] \( \textbf{Answers:} \) - (A) \( f'(x)=3x^2-12 \) - (B) Partition numbers: \( x=-2 \) and \( x=2 \) - (C) Critical numbers: \( x=-2 \) and \( x=2 \)