To find the greatest possible value of the integral from 0 to 2 of the function f(x), we can use the properties of continuous functions and integrals.
We know that the function f is continuous on the closed interval [0, 2], and we are given the condition that f(2) = 4. By the Fundamental Theorem of Calculus, the integral of f from 0 to 2 represents the area under the curve of f between these two points.
Since f(2) = 4, this means that at the point x = 2, the function value is 4. To maximize the integral, we should also consider that f(x) could possibly take on values greater than or equal to 0 for x in the interval [0, 2]. Under the assumption of continuity, we can posit that f(x) takes its maximum value (4) at that endpoint.
Assuming that f(x) is constant and equal to 4 throughout the interval [0, 2], we can compute the integral:
Integral from 0 to 2 of f(x) dx = Integral from 0 to 2 of 4 dx = 4 * (2 - 0) = 8.Thus, the greatest possible value of the integral from 0 to 2 of f(x) dx is 8.