Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \( f(x)=\frac{x^{2}+10 x+24}{x^{2}+8 x+16} \) 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. A. The function is concave upward on (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.) 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 \( x=\square \). (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no inflection points.
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To analyze the function \( f(x) = \frac{x^{2} + 10x + 24}{x^{2} + 8x + 16} \), we can start by simplifying it. The numerator can be factored to \( (x+4)(x+6) \), and the denominator is \( (x+4)^{2} \). This indicates a removable discontinuity at \( x = -4 \), where the function approaches a limit. Next, we need to find the second derivative \( f''(x) \) to determine concavity. The first derivative can be obtained using the quotient rule, and once simplified, you can differentiate again. After analysis, you will find that \( f(x) \) is concave upward on the interval \( (-\infty, -6) \cup (-4, \infty) \) and concave downward on \( (-6, -4) \). Regarding inflection points, we analyze \( f''(x) \) and solve for when it equals zero. You will discover that the inflection point occurs at \( x = -6 \). **Final Answers:** A. The function is concave upward on \( (-\infty, -6) \cup (-4, \infty) \). B. The function is concave upward on \( (-\infty, -6) \cup (-4, \infty) \). It is concave downward on \( (-6, -4) \). A. There is an inflection point at \( x = -6 \).
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