We have three factors in the function and we want to know when the function reaches a maximum in the interval [0,1].
In order to take advantage of the AM-GM Inequality, we need the sum of these factors to equal a constant. That is we need -2x rather than -x. We can get that by the following
The function reaches this maximum value for x in [0,1] if and only if
2(1 - x) = 1 + x
2 - 2x = 1 + x
1 = 3x
That is,
Is this a complete solution? Precise?