`a)3x^2-5x+3`
`=3(x^2-5/3x+1)`
`=3(x^2-2x . 5/6+25/36+11/36)`
`=3(x-5/6)^2+11/12 >= 11/12 > 0 AA x`
`=>Đpcm`
`b)2x^2+3x+4=2(x^2+3/2x+2)=2(x^2+2x. 3/4+9/16+23/16)`
`=2(x+3/4)^2+23/8 >= 23/8 > 0 AA x`
`=>Đpcm`
_________________________________________
`-5x^2+7x-3`
`=-5(x^2-7/5x+3/5)`
`=-5(x^2-2x. 7/10+49/100+11/100)`
`=-5(x-7/10)^2-11/20 <= -11/20 < 0 AA x`
`=>Đpcm`
Chứng minh bất đẳng thức luôn dương
a) \(3x^2-5x+3\)
\(=3\left(x^2-\dfrac{5}{3}x+1\right)\)
\(=3\left(x^2-2.x.\dfrac{5}{6}+\dfrac{25}{36}+\dfrac{11}{36}\right)\)
\(=3\left[\left(x-\dfrac{5}{6}\right)^2+\dfrac{11}{36}\right]\)
\(=3\left(x-\dfrac{5}{6}\right)^2+\dfrac{11}{12}\)
Do \(\left(x-\dfrac{5}{6}\right)^2\ge0\)
\(\Rightarrow3\left(x-\dfrac{5}{6}\right)^2\ge0\)
\(\Rightarrow3\left(x-\dfrac{5}{6}\right)^2+\dfrac{11}{12}>0\forall x\in R\)
Vậy \(3x^2-5x+3>0\forall x\in R\)
b) \(2x^2+3x+4\)
\(=2\left(x^2+\dfrac{3}{2}x+2\right)\)
\(=2\left(x^2+2.x.\dfrac{3}{4}+\dfrac{9}{16}+\dfrac{23}{16}\right)\)
\(=2\left[\left(x+\dfrac{3}{4}\right)^2+\dfrac{23}{16}\right]\)
\(=2\left(x+\dfrac{3}{4}\right)^2+\dfrac{23}{8}\)
Do \(\left(x+\dfrac{3}{4}\right)^2\ge0\)
\(\Rightarrow2\left(x+\dfrac{3}{4}\right)^2\ge0\)
\(\Rightarrow2\left(x+\dfrac{3}{4}\right)^2+\dfrac{23}{8}>0\forall x\in R\)
Vậy \(2x^2+3x+4>0\forall x\in R\)
Chứng minh bất đẳng thức luôn âm
a) \(-5x^2+7x-3\)
\(=-5\left(x^2-\dfrac{7}{5}x+\dfrac{3}{5}\right)\)
\(=-5\left(x^2-2.x.\dfrac{7}{10}+\dfrac{49}{100}+\dfrac{11}{100}\right)\)
\(=-5\left[\left(x-\dfrac{7}{10}\right)^2+\dfrac{11}{100}\right]\)
\(=-5\left(x-\dfrac{7}{10}\right)^2-\dfrac{11}{20}\)
Do \(\left(x-\dfrac{7}{10}\right)^2\ge0\)
\(\Rightarrow-5\left(x-\dfrac{7}{10}\right)^2\le0\)
\(\Rightarrow-5\left(x-\dfrac{7}{10}\right)^2-\dfrac{11}{20}< 0\forall x\in R\)
Vậy \(-5x^2+7x-3< 0\forall x\in R\)
