To find the greatest possible value of the integral of the function f on the closed interval from 0 to 2, given that f(x) ≤ 4 in that interval, we can apply the properties of integrals and continuous functions.
We start with the integral:
∫02 f(x) dx Since we know that the function f has an upper bound of 4, we can say:
f(x) ≤ 4 for every x in the interval [0, 2]. Therefore, we consider:
∫02 f(x) dx ≤ ∫02 4 dx Now, calculating the right-hand side:
∫02 4 dx = 4 * (2 - 0) = 8 Thus, we have:
∫02 f(x) dx ≤ 8 This tells us that the greatest possible value of the integral ∫02 f(x) dx is 8.
To achieve this maximum, the function f must equal 4 for all values of x in the interval from 0 to 2. Therefore:
f(x) = 4 for all x in [0, 2] will yield:
∫02 f(x) dx = 8 In conclusion, the greatest possible value of the integral ∫02 f(x) dx is 8.