Question

Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 4x3 − 6x2 − 24x + 1, [−2, 3] absolute minimum value absolute maximum value

Answer 1

When looking for extremes on an interval, one must examine the turning points and the values of the function at the ends of the interval. Here the turning points are where the derivative is zero:. j

... 12x² -12x -24 = 0

... 12(x -2)(x +1) = 0

Since the turning points, x=-1, x=2, are in the interval we have four function values to compute.

... f(-2) = ((4(-2) -6)(-2) -24)(-2) +1 = -7

... f(-1) = ((4(-1) -6)(-1) -24)(-1) +1 = 15

... f(2) = ((4(2) -6)(2) -24)(2) +1 = -39

... f(3) = ((4(3) -6)(3) -24)(3) +1 = -17

This shows the extremes to be at the turning points.

The absolute minimum value is -39 at x=2.

The absolute maximum value is 15 at x=-1.