Question

Find the absolute maximum and absolute minimum for f (x )equals x cubed minus 2 x squared minus 4 x plus 2 on the interval 0 less or equal than x less or equal than 3.

Answer 1

9514 1404 393

Answer:

• maximum: 2
• minimum: -6

Explanation:

The extrema will be at the ends of the interval or at a critical point within the interval.

The derivative of the function is ...

f'(x) = 3x² -4x -4 = (x -2)(3x +2)

It is zero at x=-2/3 and at x=2. Only the latter critical point is in the interval. Since the leading coefficient of this cubic is positive, the right-most critical point is a local minimum. The coordinates of interest in this interval are ...

f(0) = 2

f(2) = ((2 -2)(2) -4)(2) +2 = -8 +2 = -6

f(3) = ((3 -2)(3) -4)(3) +2 = -3 +2 = -1

The absolute maximum on the interval is f(0) = 2.

The absolute minimum on the interval is f(2) = -6.