Final answer:
To find the absolute extrema of the function 2x³ - 3x² - 12x + 2 on the interval [-5,5], calculate the derivative, find and evaluate the critical points, and compare these values with the function values at the endpoints of the interval to determine the absolute maximum and minimum.
Step-by-step explanation:
To find the absolute extrema of the function f(x) = 2x³ - 3x² - 12x + 2 on the interval [-5,5], we need to follow these steps:
- Calculate the derivative of the function, f'(x) = 6x² - 6x - 12.
- Find the critical points by setting the derivative equal to zero and solving for x.
- Evaluate the function at the critical points and at the endpoints of the interval, x = -5 and x = 5.
- Compare all these values to determine which is the highest (absolute maximum) and which is the lowest (absolute minimum).
Finding the critical points involves solving the equation f'(x) = 6x² - 6x - 12 = 0. This can be factored or solved using the quadratic formula. Once the critical points are found, plug them back into the original function to get their corresponding y-values.
After evaluating the critical points and endpoints, the largest of these y-values will be the absolute maximum and the smallest will be the absolute minimum of the function on the given interval.