Find the intervals in which the function f(x) = x⁴ - x³ - 5x² + 24x + 12 is (a) strictly increasing, (b) strictly decreasing.

We have, f(x) = x⁴ - x³ - 5x² + 24x + 12 ⇒ f'(x) = 4x³ - 3x² - 10x + 24 ⇒ f''(x) = 12x² - 6x - 10
If f''(x) > 0 the function is strictly decreasing and if f''(x) < 0 ⇒ 12x² - 6x - 10 = 0 ⇒ x = (6 ± √36 + 4 × 3 × 10)/6 = (6 ± √156)/6 = (6 ± 12.48996)/6 ⇒ x = 3.08 or -1.08
We get that f''(x) < 0 when -1.08 < x < 3.08 and f''(x) > 0 when x > 3.08 and x < -1.08
∴ f(x) is strictly increasing when -1.08 < x < 3.08 and f(x) is strictly decreasing when x > 3.08 and x < -1.08