Final answer:
To find the absolute extrema of the function f(x) = x² - 2x within the interval [-1, 2], we find the critical points and evaluate the function at those points and the endpoints of the interval. The absolute maximum value is 3 and the absolute minimum value is -1.
Step-by-step explanation:
To find the absolute extrema of a function, we need to find the highest and lowest points of the function within a given interval. In this case, the function is f(x) = x² - 2x and the interval is [-1, 2].
To find the absolute extrema, we need to evaluate the function at the critical points and the endpoints of the interval.
First, let's find the critical points by taking the derivative of the function and setting it equal to zero:
f'(x) = 2x - 2 = 0
Solving this equation, we find x = 1 as the critical point.
Next, we evaluate the function at the critical point and the endpoints of the interval:
f(-1) = (-1)² - 2(-1) = 3
f(1) = (1)² - 2(1) = -1
f(2) = (2)² - 2(2) = 0
The highest point is f(-1) = 3 and the lowest point is f(1) = -1 within the interval [-1, 2]. Therefore, the absolute maximum value is 3 and the absolute minimum value is -1.