Final Answer:
The local minimum value of the function f(x) = 3x⁴ - 8x³ - 144x² + 6 is -∞, and the local maximum value is 6.
Step-by-step explanation:
To find the local minimum and maximum values of the given function, we first need to find its critical points. We can do this by taking the derivative of the function and setting it equal to zero to solve for x. The derivative of f(x) = 3x⁴ - 8x³ - 144x² + 6 is f’(x) = 12x³ - 24x² - 288x. Setting f’(x) = 0, we get critical points at x = -6, x = 0, and x = 6.
Next, we use the second derivative test to determine whether these critical points correspond to local minima or maxima. Evaluating f’’(x) at each critical point, we find that f’’(-6) = -432, f’’(0) = 0, and f’’(6) = 432. Since f’’(-6) < 0, f’’(0) = 0, and f’’(6) > 0, we conclude that there is a local maximum at x = -6 and a local minimum at x = 6.
Substituting these critical points back into the original function, we find that f(-6) = 582, f(0) = 6, and f(6) = -582. Therefore, the local minimum value is -∞ (as x approaches positive infinity), and the local maximum value is 6.