Final answer:
The critical numbers of the given function are x = 0, x = 2, and x = -1. x = 0 is neither a max or min, x = 2 is a local min, and x = -1 is a local max.
Step-by-step explanation:
To find the critical numbers of the function f(x) = 3x⁵- 15x⁴ + 20x³ - 7, we need to find the values of x where the derivative of the function is equal to zero or does not exist. The derivative of f(x) is f'(x) = 15x⁴ - 60x³ + 60x². Setting f'(x) = 0 and solving for x, we get x = 0, x = 2, and x = -1. These are the critical numbers.
To classify the critical numbers, we can use the second derivative test. The second derivative of f(x) is f''(x) = 60x³ - 180x² + 120x. Evaluating f''(x) at each critical number, we have f''(0) = 0, f''(2) = 96, and f''(-1) = -24. Since f''(0) = 0, this critical number does not help us determine the nature of the extremum. However, since f''(2) > 0, x = 2 corresponds to a local minima, and since f''(-1) < 0, x = -1 corresponds to a local maxima.
Therefore, rounding the answers to three decimal places, we have:
x = 0 is neither a max or min.
x = 2 is a local min.
x = -1 is a local max.