Final answer:
The concavity of the graph of the function f(x) = x(x-4)³ can be determined by analyzing the second derivative of the function. The function is concave upward in the intervals (0, 4) and (4, ∞), and concave downward in the interval (-∞, 0).
Step-by-step explanation:
The concavity of the graph of the function f(x) = x(x-4)³ can be determined by analyzing the second derivative of the function. The function is concave upward in the intervals where the second derivative is positive, and concave downward in the intervals where the second derivative is negative.
To find the second derivative, we need to differentiate the function twice.
First, we differentiate f(x) with respect to x: f'(x) = 4x(x-4)² + x(x-4)³'.
Simplifying this gives f'(x) = 4x(x-4)² + x(x-4)²(3).
Next, we differentiate f'(x) with respect to x: f''(x) = (4x(x-4)²)' + ((4x(x-4)²)'(3) + x(x-4)²(3)'.
Simplifying this gives f''(x) = 12x(x-4) + 4(x-4)² + (3x(x-4)² + x(x-4)²'(3)).
Simplifying further, we get f''(x) = 12x(x-4) + 4(x-4)² + (3x(x-4)² + x(x-4)²(3)).
By analyzing the sign of the second derivative in different intervals of x, we can determine the concavity of the graph.
The open intervals on which the graph is concave upward are (0, 4) and (4, ∞), and the open interval on which the graph is concave downward is (-∞, 0).