Question Detail

To find the maximum value, we first find the first derivative of the function and set it to zero to find critical points. Given f(x) = x³ - 27x + 15.First derivative: f'(x) = 3x² - 27.Set f'(x) = 0: 3x² - 27 = 0 => 3x² = 27 => x² = 9 => x = ±3.Next, we use the second derivative test to determine if these points are local maxima or minima.Second derivative: f''(x) = 6x.Evaluate 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.Thus, the maximum value occurs at x = -3.