Summarize the pertinent information obtained by applying the \( f(x)=(x-4)\left(x^{2}-8 x-32\right) \) Select the correct choice below and, if necessary, fill in the a A. The function \( f \) has a local maximurn at \( x-\square \). (Use a comma to separate answers as needed.) B. The function \( f \) has no local maximum. Select the cortect choice below and, if necessary, fill in the A. The lunction \( f \) has a local mirimum at \( x=\square \). (Use a comma to separale answers as needed.) B. The function \( f \) has no local minimum

We begin with the function \[ f(x)=(x-4)\left(x^{2}-8x-32\right). \] It is easier to work with the fully expanded form. Expanding, we have \[ \begin{aligned} f(x) &= x\left(x^{2}-8x-32\right) - 4\left(x^{2}-8x-32\right) \\ &= x^3 - 8x^2 -32x - 4x^2 +32x +128 \\ &= x^3 - 12x^2 + 128. \end{aligned} \] Next, we compute the first derivative to find potential local extrema: \[ f'(x) = \frac{d}{dx}\left(x^3 - 12x^2 + 128\right) = 3x^2 - 24x. \] Factor the derivative: \[ f'(x)= 3x(x-8). \] Setting \( f'(x)=0 \) gives the critical points: \[ 3x(x-8)=0 \quad \Longrightarrow \quad x=0 \quad \text{or} \quad x=8. \] To determine the nature of these critical points, we compute the second derivative: \[ f''(x)= \frac{d}{dx}\left(3x^2-24x\right)= 6x-24 = 6(x-4). \] Evaluate \( f''(x) \) at each critical point: \[ \begin{aligned} f''(0) &= 6(0-4)= -24 \quad (\text{negative: local maximum}), \\ f''(8) &= 6(8-4)= 24 \quad (\text{positive: local minimum}). \end{aligned} \] Thus, we have: - A local maximum at \( x=0 \). - A local minimum at \( x=8 \). The choices provided indicate: For the maximum: - Option A: The function \( f \) has a local maximum at \( x=\boxed{0} \). - Option B: The function \( f \) has no local maximum. For the minimum: - Option A: The function \( f \) has a local minimum at \( x=\boxed{8} \). - Option B: The function \( f \) has no local minimum. The correct selections are Option A for both questions: - The function \( f \) has a local maximum at \( x=0 \). - The function \( f \) has a local minimum at \( x=8 \).