For the function below, find \( (A) f^{\prime}(x) \), (B) the partition numbers lor \( \mathrm{f}^{\prime} \), (C) the critical numbers of f . \( f(x)=\frac{8}{x+3} \) (A) \( f^{\prime}(x)=\square \)

\( f(x)=\frac{8}{x+3}=8(x+3)^{-1} \) **(A) Finding \( f^{\prime}(x) \):** Differentiate using the power rule and chain rule: \[ f^{\prime}(x)=8\cdot(-1)(x+3)^{-2}\cdot(1)=-\frac{8}{(x+3)^2} \] **(B) The partition numbers for \( f^{\prime}(x) \):** The function \( f^{\prime}(x)=-\frac{8}{(x+3)^2} \) is undefined when the denominator is zero. Solve: \[ (x+3)^2=0 \quad\Longrightarrow\quad x=-3 \] Thus, the domain of \( f^{\prime}(x) \) is split by the partition number \( x=-3 \). The partition numbers for \( f^{\prime} \) are at \( x=-3 \). **(C) The critical numbers of \( f \):** Critical numbers occur where \( f^{\prime}(x)=0 \) or \( f^{\prime}(x) \) is undefined, provided \( x \) is in the domain of \( f \). 1. Set \( f^{\prime}(x)=0 \): \[ -\frac{8}{(x+3)^2}=0 \] The numerator \(-8\) is never zero, so there is no solution from \( f^{\prime}(x)=0 \). 2. Points where \( f^{\prime}(x) \) is undefined: \( f^{\prime}(x) \) is undefined at \( x=-3 \). However, check the domain of \( f(x)=\frac{8}{x+3} \): It is also undefined at \( x=-3 \). Since \( x=-3 \) is not in the domain of \( f \), it is not considered a critical number. Thus, \( f \) has no critical numbers.