Obtain a maximum value of the function \(f(x)=x^3-12x+11\).

To obtain the maximum value of the given function f(x) = x^3 - 12x + 11, we need to find its critical points. The critical points of a function are the points where the derivative of the function is either zero or undefined. So, let's find the derivative of the given function: f'(x) = 3x^2 - 12 Now, we'll set f'(x) = 0 and solve for x: 3x^2 - 12 = 0 x^2 - 4 = 0 (x - 2)(x + 2) = 0 So, the critical points of the function are x = 2 and x = -2. To determine whether these are maximum or minimum points, we'll take the second derivative of the function: f''(x) = 6x Now, we'll substitute the critical points into the second derivative: f''(2) = 12 f''(-2) = -12 Since f''(2) is positive, the critical point x = 2 is a minimum point. Since f''(-2) is negative, the critical point x = -2 is a maximum point. Therefore, the maximum value of the given function is obtained when x = -2. Now, we'll substitute x = -2 into the original function: f(-2) = (-2)^3 - 12(-2) + 11 f(-2) = -8 + 24 + 11 f(-2) = 27 So, the maximum value of the given function is 27, and the correct option is (D).