Question

CMR: \(x-x^2-1< 0\forall x\)

Answer 1

\(x-x^2-1\)

\(=-\left(x^2-x+1\right)\)

\(=-\left[\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{1}{4}-1\right]\)

\(=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{5}{4}\)

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 \(x-x^2-1< 0\forall x\)