Answer:
Explanation:
You want the absolute extremes of the function f(x) = x³ -3x² +1 on the closed interval [-1/2, 4].
Extreme values
The extreme values will be either the turning points of the function or the ends of the interval. We must examine all of those. A graphing calculator is helpful for evaluating the function at various points.
Interval ends
f(-1/2) = 1/8
f(4) = 17
Turning points
The turning points are found where the derivative is zero.
f'(x) = 3x² -6x = 3x(x -2)
This function is zero where the factors are 0, at x=0 and x=2.
f(0) = 1
f(2) = -3
The maximum value is found at the right end of the interval: f(4) = 17.
The minimum value is found at the right turning point: f(2) = -3.