Maximums and Minimums

Question

The maximum value of f(x) = x³ - 3x² - 9x + 2 on the interval [0, 6] is

Options:

1. 2
2. 56
3. 135
4. There is no maximum on [0,6]

Answer 1

f(x) = x³ - 3x² - 9x + 2

To find the maxima and minima, take the 1st derivative and set = to 0:

f'(x) = 3x² - 6x - 9 = 0

x² - 2x - 3 = 0

x = 2, -1

Only x = 2 falls in the specified interval. Solve f(x) for x = 2:

8 - 12 - 18 + 2 = -20 This appears to be a local minimum. It appears that the maximum will be at x = 6:

f(6) = 216 - 108 - 54 + 2 = 56

CHECK by graphing the function using a graphing calculator or online tool.