Answer:
- minima: (-21, -37044) and (42, -37044)
- maxima: (0, 0) and (63, 0)
Explanation:
You want the absolute extreme values of f(x) = x³ -63x² on the interval [-21, 63].
Extremes
The absolute extremes will be located at the ends of the interval and/or at places within the interval where the derivative is zero.
Derivative
The derivative of f(x) is ...
f'(x) = 3x² -126x
This is zero when its factors are zero.
f'(x) = 0 = 3x(x -42)
x = {0, 42} . . . . . . . . . within the interval [-21, 63]
Function values
The attachment shows the function values at these points and at the ends of the interval. It tells us the minima are located at x=-21 and x=42. The maxima are located at x=0 and x=63. Their values are -37044 and 0, respectively.
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Additional comment
These are absolute extrema in the interval because no other values are larger than these maxima or smaller than the minima.
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