Final answer:
The function f(x)=(x−6)²/³ is increasing on the interval (6, 20] and decreasing on the interval [0, 6).
Step-by-step explanation:
To find the intervals where the function f(x)=(x−6)²/³ is increasing or decreasing, we first need to determine its derivative to identify any critical points and analyze its slope. The derivative of (x-6)²/³, with respect to x, can be calculated using the chain rule.
Let's denote g(x) = x - 6 and h(g) = g²/³. By applying the chain rule, the derivative of f(x) with respect to x is f'(x) = (2/3)g²/³-1 * g'(x), which simplifies to f'(x) = (2/3)(x-6)−³/³. Since the derivative only involves x in the numerator, it is positive for x > 6 and negative for x < 6, implying that the function is increasing for x > 6 and decreasing for x < 6.
The interval of increase is therefore (6, ∞) and the interval of decrease is (-∞, 6). Remember, however, that we should restrict our analysis to x values for which the function is defined. If x must be a real number between 0 and 20, including the endpoints, then the function increases on (6, 20] and decreases on [0, 6).