The quadratic function we have is f(x) = -2x^2 + 2x
To determine the extreme point of a function f(x), solve f'(x) = 0. Substitute the solution(s) xs in f''(x). If f''(xs) is negative we have a maximum value at x = xs , else if f''(xs) is positive, we have a minimum value at x = xs.
f(x) = -2x^2 + 2x
f'(x) = -4x + 2
-4x + 2 = 0
=> x = 1/2
f''(x) = -4 which is always negative.
The value at x = 1/2 is a maximum value. This is equal to f(1/2) = -2*(1/2)^2 + 2*(1/2) = -1/2 + 1 = 1/2.
The maximum value of the given function is f(1/2) = 1/2