\(x-x^2-1\)
\(=-\left(x^2-x+1\right)\)
\(=-\left[\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{1}{4}-1\right]\)
\(=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{5}{4}\)
Ta có :
\(-\left(x-\dfrac{1}{2}\right)^2\le0\Rightarrow-\left(x-\dfrac{1}{2}\right)^2+\dfrac{5}{4}\le\dfrac{5}{4}< 0\forall x\)
hay \(x-x^2-1< 0\forall x\)
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