Functionally impossible.

Calculus Level 5

\(f(x)\) is defined for \(x\ge0\) and has a continuous derivative. It satisfies \(f(0)=1, f^{'}(0)=0\) and \([1+f(x)]f^{''}(x)=1+x\). Let \(f(1)= A\), then the impossible values of \(A\) is/are:

Comment:

I worked really hard to get \(Lv.5\) in calculus, just so that I could post this question. Please also give solution. Much thanks.

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