Need help of max and min calculus, thank you.

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

The function is a cubic function with a positive leading coefficient. Therefore, it will be increasing until the first critical point (local maximum) and after the second critical point (local minimum).

Let's find the x values for the local extrema:

f'(x) = 3x² - 6x - 9

3x² - 6x - 9 = 0

3(x² - 2x - 3) = 0

3(x - 3)(x + 1) = 0

x = -1, x = 3

Therefore, f(x) is increasing on (-∞,-1) and (3,∞) and the maximum on [0,6] will be at x = 6.

f(6) = 56