Question Detail
Question
The function y = 9x³ - 45x² + 48x + 11 has a minimum value at y =?
Correct Answer
Option B is the correct answer.
Detailed Explanation
From the analysis in Q62, the critical points are x = 8/3 and x = 2/3. The second derivative test showed that x = 8/3 corresponds to a local minimum (d²y/dx² = 54 > 0).To find the minimum 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 + 128 + 11y = 512/3 - 320 + 128 + 11y = 512/3 - 181 = 512/3 - 543/3 = -31/3.The calculated minimum value is -31/3. Options B and D are both -7/3. There is a discrepancy between the calculated value and the options provided in the document. The document highlights option B, -7/3, which is not the mathematically correct minimum value for the given function.
Hint
After finding the critical points, use the second derivative test (d²y/dx²) to identify which point corresponds to a local minimum. Then substitute that x-value into the original function.