Final answer:
To find the relative extrema of the function f(x) = x³ - 12x - 3, we set the derivative equal to zero to find the critical points. Then, we evaluate the second derivative at the critical points to determine if they are local maxima or minima.
Step-by-step explanation:
To find the relative extrema of a function, we first need to find the critical points by setting the derivative of the function equal to zero.
The derivative of f(x) = x³ - 12x - 3 is f'(x) = 3x² - 12. Setting f'(x) = 0, we get 3x² - 12 = 0. Solving this equation gives us x = ±2.
Next, we need to determine whether these critical points are local maxima or minima by evaluating the second derivative of the function f''(x). If f''(x) > 0, the critical point is a local minimum, and if f''(x) < 0, the critical point is a local maximum.
Taking the second derivative, f''(x) = 6x. Evaluating f''(x) at x = ±2, we find that f''(±2) = ±12. Since f''(-2) < 0 and f''(2) > 0, we can conclude that x = -2 is a local maximum, and x = 2 is a local minimum.