Summarize the pertinent information obtained by a \( f(x)=(x-4)\left(x^{2}-8 x-32\right) \) Select the correct choice below and, if necessary, A. The function \( f \) has an inflection point at \( x= \) (Use a comma to separate answers as ne B. The function if has no inflection point. Choose the correct graph of \( y=f(x) \) below.

To find the inflection point of the function \( f(x) = (x-4)(x^2-8x-32) \), we need to find the second derivative of the function and set it equal to zero. First, let's expand the function: \[ f(x) = (x-4)(x^2-8x-32) \] \[ f(x) = x^3 - 8x^2 - 32x - 4x^2 + 32x + 128 \] \[ f(x) = x^3 - 12x^2 + 128 \] Now, let's find the first derivative of the function: \[ f'(x) = 3x^2 - 24x \] Next, let's find the second derivative of the function: \[ f''(x) = 6x - 24 \] To find the inflection point, we set the second derivative equal to zero: \[ 6x - 24 = 0 \] \[ 6x = 24 \] \[ x = 4 \] Therefore, the function \( f \) has an inflection point at \( x = 4 \). Now, let's analyze the graph of \( y = f(x) \) to determine if it has any inflection points. The graph of \( y = f(x) \) is a cubic function with a positive leading coefficient, which means it opens upwards. The inflection point at \( x = 4 \) indicates that the graph changes from concave down to concave up at this point. Therefore, the correct graph of \( y = f(x) \) is the one that shows the inflection point at \( x = 4 \).