Final answer:
The local maximum value of the cubic function f(x) = x³ - 9x² - 21x occurs at x = -1, and the local minimum value occurs at x = 7, determined by the first and second derivatives of the function.
Step-by-step explanation:
To find the local minimum and maximum values of the cubic function f(x) = x³ - 9x² - 21x, we need to find the first and second derivatives of f(x). The first derivative, f'(x), will tell us where the function has critical points, which occur where f'(x) is equal to zero or is undefined. These points are potential candidates for local minima or maxima. The second derivative, f''(x), can help us determine if the critical points are indeed local minima or maxima.
First, let's find the first derivative:
f'(x) = 3x² - 18x - 21. Set this equal to zero to find the critical points: 3x² - 18x - 21 = 0. Factoring, we get (3x + 3)(x - 7) = 0. Thus, the critical points are x = -1 and x = 7.
Next, we'll find the second derivative: f''(x) = 6x - 18. Evaluate f''(x) at the critical points to determine their nature. At x = -1, f''(-1) = -24, which means x = -1 is a local maximum because the second derivative is negative. At x = 7, f''(7) = 24, which suggests x = 7 is a local minimum because the second derivative is positive.
Therefore, the local maximum value of the function occurs at x = -1, and the local minimum value occurs at x = 7.