Question Detail
Question
The function y = x3 + 6x2 - 15x + 5 has a minimum value of y = ?
Correct Answer
Option B is the correct answer.
Detailed Explanation
From the previous question (Question 31), we found the critical points of y = x3 + 6x2 - 15x + 5 by setting the first derivative to zero:y' = 3x2 + 12x - 15 = 3(x+5)(x-1) = 0The critical points are x = -5 and x = 1.Using the second derivative test (y'' = 6x + 12):At x = -5: y''(-5) = -18 (local maximum)At x = 1: y''(1) = 18 (local minimum)To find the minimum value of y, substitute x = 1 into the original function:y(1) = (1)3 + 6(1)2 - 15(1) + 5y(1) = 1 + 6 - 15 + 5y(1) = 7 - 15 + 5y(1) = -8 + 5 = -3.The minimum value of y is -3.
Hint
Recall the critical points found from the first derivative. Identify which one corresponds to a minimum using the second derivative test, then evaluate the original function at that point.