Final answer:
The graph of the function f(x) = 10x² - 10sin(2x) is concave upward on the interval [0,π], as determined by its second derivative, which is always positive over this interval.
Step-by-step explanation:
To describe the concavity of the graph of the function f(x) = 10x² - 10sin(2x) on the interval [0,π], we first need to find the second derivative of the function. The concavity of a graph is determined by the sign of its second derivative. If the second derivative is positive on an interval, the graph is concave upward on that interval. If it is negative, the graph is concave downward.
To find the second derivative, start with the first derivative f'(x) = 20x - 20cos(2x). Then, derive it again to get the second derivative f''(x) = 20 + 40sin(2x). Since the max value of sin(2x) is 1, the smallest value for f''(x) is 20 - 40, which is still positive. This means that, on the interval from 0 to π, the second derivative is always positive. Therefore, the graph of f(x) is concave upward over the entire interval [0,π].