Answer:
(a) maximum = 400; minimum = 8
(b) maximum = 348; minimum = -100
(c) maximum = 400; minimum = -100
Step-by-step explanation:
The absolute extrema will be either at the ends of the interval or at a turning point in the interval. Here, the turning points can be found from the derivative:
f'(x) = 3x^2 +12x -63 = 3(x^2 +4x -21)
f'(x) = 3(x +7)(x -3)
The derivative is zero at x=-7 and at x=3. Since the cubic has a positive leading coefficient, the extreme at x=-7 is a maximum; that at x=3 is a minimum.
So, the values we are concerned with are ...
- f(-8) = 384
- f(-7) = 400 . . . turning point
- f(-5) = 348
- f(0) = 8
- f(3) = -100 . . . turning point
- f(4) = -84
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(a)
1. The left turning point is in the interval, so the absolute maximum is f(-7) = 400.
2. The absolute minimum is at the right end of the interval, at x=0. Its value is f(0) = 8.
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(b)
1. The absolute maximum is at the left end of the interval. Its value is f(-5) = 348.
2. The absolute minimum is at the right turning point: f(3) = -100.
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(c)
1. The absolute maximum is at the left turning point: f(-7) = 400.
2. The absolute minimum is at the right turning point: f(3) = -100.
