Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). Points: 0 of 1 \[ f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} \] A. There is a local maximum at \( x= \) \( \square \) There is no local minimum. (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is a local minimum at \( x= \) \( \square \) There is no local maximum. (Type an integer or a decimal. Use a comma to separate answers as needed.) C. There is a local maximum at \( x= \) \( \square \) and there is a local minimum at \( x= \) \( \square \) (Type integers or decimals. Use a comma to separate answers as needed.) D. There are no local extrema. ind 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. A. The function is concave upward on \( \square \) It is never concave downward. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.) B. The function is concave upward on \( \square \) It is concave downward on \( \square \) ]. (Type exact answers. Type your answers in interval notation. Use commas to separate answers as needed.) C. The function is concave downward on \( \square \) It is never concave upward. (Type an exact answer. Type your answer in interval notation. Use a comma to separate answers as needed.)

To analyze the function \( f(x) = \frac{x^2 + 10x + 24}{x^2 + 8x + 16} \), we first need to determine its local extrema. We can start by finding the critical points through the first derivative test. Upon calculating the first derivative and setting it to zero, we can find potential local maximum and minimum values. By applying the second derivative test, we can classify the critical points. For this function, upon performing the necessary calculations, we discover that there is a local maximum at \( x = -2 \) and a local minimum at \( x = -4 \). In terms of concavity, we need to analyze the second derivative of \( f(x) \). Upon calculation, we can determine that \( f(x) \) is concave upward on the intervals \( (-\infty, -4) \) and \( (-2, \infty) \), while it is concave downward on the interval \( (-4, -2) \). Thus, the summarized answers to your queries are: A. There is a local maximum at \( x = -2 \). There is no local minimum. B. There is a local minimum at \( x = -4 \). There is no local maximum. C. There is a local maximum at \( x = -2 \) and there is a local minimum at \( x = -4 \). D. There are no local extrema. For concavity: A. The function is concave upward on \( (-\infty, -4) \cup (-2, \infty) \). It is never concave downward. B. The function is concave upward on \( (-\infty, -4) \cup (-2, \infty) \). It is concave downward on \( (-4, -2) \). C. The function is concave downward on \( (-4, -2) \). It is never concave upward.