Question

Cm 2x-2x²-1<0 với mọi số thực x

Answer 1

\(2x-2x^2-1\)

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

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

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

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

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

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

vậy \(2x-2x^2-1< 0\) với mọi số thực x