Question Detail
Question
The function y = 9x³ – 45x² + 48x + 11 has a minimum value at y = ?
Correct Answer
Option D is the correct answer.
Detailed Explanation
From the analysis in the previous question (Q62):Critical points are x = 8/3 (local minimum) and x = 2/3 (local maximum).To find the minimum y-value, substitute x = 8/3 into the original function:y = 9(8/3)³ – 45(8/3)² + 48(8/3) + 11y = 9(512/27) – 45(64/9) + 128 + 11y = 512/3 – 5 * 64 + 139y = 512/3 – 320 + 139y = 512/3 – 181 = 512/3 – 543/3 = -31/3.The calculated minimum value is -31/3. However, the provided correct option is D, which is 7/3. There is a discrepancy between the calculated value and the option provided.
Hint
After finding the critical points and identifying the one corresponding to a minimum using the second derivative test, substitute that x-value into the original function y(x) to find the minimum y-value.