Question

[tran hang]

^{c/m voi moi gia tri cua x ta lon co}

^{a,x^4+3x^2+3 lon hon 0}

^{b,(x^2+2x+3)(x^2+2x+4)+3 lon hon 0}

Answer 1

[Lê Song Phương]

a) Xét biểu thức \(x^4+3x^2+3\)

Ta thấy \(x^4\ge0\forall x\inℝ;3x^2\ge0\forall x\inℝ\)và 3 > 0

\(\Rightarrow x^4+3x^2+3>0\forall x\inℝ\)

b) Ta thấy \(x^2+2x+3=\left(x^2+2x+1\right)+2=\left(x+1\right)^2+2\)

Vì \(\left(x+1\right)^2\ge0\Leftrightarrow\left(x+1\right)^2+2\ge2>0\Leftrightarrow x^2+2x+3>0\)(1)

\(\Leftrightarrow x^2+2x+4>1>0\)(2)

Mà 3 > 0 (3)

Từ (1), (2) và (3) \(\Rightarrow\left(x^2+2x+3\right)\left(x^2+2x+4\right)+3>0\forall x\inℝ\)