Final answer:
The function f(x) = √x(x-2)^²/3 involves analyzing critical points, domain, and behavior patterns to sketch its graph. The function is defined for x ≥ 0 with a local minimum at x = 2, and it increases on either side of this point.
Step-by-step explanation:
The question asks for an analysis of the function f(x) = √x(x-2)^²/3 and how to sketch its graph. To analyze the function, we should consider its domain, potential asymptotes, intercepts, and behavior at critical points such as x = 0 and x = 2. Since we are dealing with a square root function, the domain of the function is x ≥ 0. The behavior of the function changes at x = 2 because of the square term. To sketch the graph, start by identifying the y-intercept at x = 0 and note that the function has a critical point at x = 2 where the behavior of the function changes. The graph will start from the origin, as the square root of 0 is 0, and will increase until x = 2. Due to the (x-2)^² term, the function has a local minimum at x = 2. For values of x > 2, the function increases due to the square of (x-2).
To represent the graph accurately, label the graph with f(x) and x, and scale the x and y axes. Make sure to account for the function being undefined for x < 0 and include the point where the function changes direction at x = 2.