Final answer:
To find the relative extrema of f(x) = x^3 - 3x^2, the first and second derivatives are used to identify critical points and determine whether they are relative minima or maxima. The function has a relative maximum at x = 0 and a relative minimum at x = 2.
Step-by-step explanation:
The question involves finding the relative extrema of the function f(x) = x^3 - 3x^2.
To find the relative extrema, we first calculate the derivative f'(x) to find the critical points, where f'(x) = 0 or undefined. For this function, the derivative f'(x) = 3x^2 - 6x.
Setting this equal to zero gives us the critical points x = 0 and x = 2.
Next, we use the second derivative test to determine the nature of each critical point by calculating f''(x) = 6x - 6. If f''(x) is positive at a critical point, that point is a relative minimum;
if f''(x) is negative, we have a relative maximum. At x = 0, since f''(0) = -6, we have a relative maximum. At x = 2, since f''(2) = 6, we have a relative minimum.