Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of \( f(x)=\frac{8 x}{x^{2}-4} \) Find the intervals where \( f(x) \) is concave upward or downward. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. The function is concave upward on It is never concave downward. B. The function is concave downward on . It is never concave upward C. The function is concave upward on . It is concave downward on Find the location of any inflection points of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. There is an inflection point at \( x= \) B. There are no inflection points.

To sketch the graph of \( f(x)=\frac{8x}{x^{2}-4} \), start by identifying vertical asymptotes where the denominator equals zero, \( x^{2} - 4 = 0 \) gives \( x = 2 \) and \( x = -2 \). The horizontal asymptote is determined by examining the behavior as \( x \to \infty \) and \( x \to -\infty \), which approaches \( y=0 \). Plot intercepts, \( f(0)=0 \) (y-intercept) and \( f(x) \) at \( x = 2 \) and \( x = -2 \). This creates a detailed graph that highlights the function's behavior across intervals. For concavity, compute the second derivative \( f''(x) \). The function is concave upward where \( f''(x) > 0 \) and concave downward where \( f''(x) < 0 \). This analysis typically leads to specific intervals. After completing the calculations, you’ll find that the function is concave upward on intervals such as \( (-\infty, -2) \cup (2, \infty) \) and concave downward on intervals such as \( (-2, 2) \). For inflection points, they occur where \( f''(x) \) changes sign. After solving \( f''(x) = 0 \) and verifying with the sign test, you may find inflection points, such as at \( x = 0 \), indicating where the concavity of the function changes.