Question

CMR 2m^4 + 2m + 1 >= 0 với mọi m

Answer 1

ta có \(2m^4+2m+1\)

\(=2m^4-2m^2+\dfrac{1}{2}+2m^2+2m+\dfrac{1}{2}\)

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

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

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

\(=2\left(m^2-\dfrac{1}{2}\right)^2+2\left(m+\dfrac{1}{2}\right)^2\ge\forall m\) ( đpcm)