Question Detail

To find the maximum value of f(x) = x³ - 27x + 15, we first find the first derivative and set it to zero to find critical points.f'(x) = 3x² - 27Setting f'(x) = 0:3x² - 27 = 03x² = 27x² = 9x = ±3Next, we find the second derivative to determine if these points are maxima or minima.f''(x) = 6xEvaluate f''(x) at x = 3:f''(3) = 6(3) = 18. Since f''(3) > 0, x = 3 is a local minimum.Evaluate f''(x) at x = -3:f''(-3) = 6(-3) = -18. Since f''(-3) < 0, x = -3 is a local maximum.Therefore, the maximum value occurs at x = -3.