Solution with AM-GM Inequality

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?

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