x-x2-1
↔ -(x2-x+1)
↔ -(x2-2✖ ½x+¼+¾)
↔- [(x-½)2+¾]↔- (x-½)2 -¾
vì (x-½)2>=0 vs mọi x nên -(x-½)2-¾ <=¾ vs mọi x hay -(x-½)2-¾ <0 vs mọi x
vậy x-x2-1<0 vs mọi x
Ta có : x-x2-1=
-x2+x-1/4-3/4=-(x2-x+(1/2)2)-3/4=-(x-1/2)2-3/4
Vì -(x-1/2)2 \(\le\)0 (\(\forall x\in R\))
=>-(x-1/2)2-3/4\(\le\)-3/4 (\(\forall x\in R\))
=>-(x-1/2)2-3/4<0 (\(\forall\)x\(\in\)R)
=>ĐPCM
Ta có x-x2-1 = - (x2+x+1)=-(x2+2.\(\dfrac{1}{2}\)x+ (\(\dfrac{1}{2}\))2-\(\dfrac{1}{4}\)+1)= -( x+\(\dfrac{1}{2}\))2 -\(\dfrac{3}{4}\)
Ta có -( x+\(\dfrac{1}{2}\))2 \(\le\) 0 với mọi x
\(\Rightarrow\) -(x+\(\dfrac{1}{2}\))2 -\(\dfrac{3}{4}\) \(\le\) \(\dfrac{-3}{4}\) < 0
Vậy ............
