Question

Bài 24: Chứng minh các đa thức sau luôn dương với mọi x,y

a) x²+ 2x +2

b) 4x² - 12x +11

c) x² - x +1

Answer 1

a/ \(x^2+2x+2=x^2+2x+1+1=\left(x+1\right)^2+1\)

vì: \(\left(x+1\right)^2\ge0\forall x\Rightarrow\left(x+1\right)^2+1\ge1>0\left(đpcm\right)\)

b/ \(4x^2-12x+11=\left(4x^2-2\cdot2x\cdot3+9\right)+2=\left(2x-3\right)^2+2\)

vì: \(\left(2x-3\right)^2\ge0\forall x\Rightarrow\left(2x-3\right)^2+2\ge2>0\left(đpcm\right)\)

c/ \(x^2-x+1=x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\)

Vì: \(\left(x-\dfrac{1}{2}\right)^2\ge0\forall x\Rightarrow\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\left(đpcm\right)\)