Question

Chứng minh rằng với mọi giá trị x, ta có:

a ) \(x^2+4x+5>0\)

b ) \(-x^4+4x^2-7< 0\)

Answer 1

Câu a :

\(x^2+4x+5\)

\(=\left(x^2+4x+4\right)+1\)

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

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

Vậy \(x^2+4x+5>0\left(\forall x\right)\)

Câu b :

\(-x^4+4x^2-7\)

\(=\left(-x^4+4x^2-4\right)-3\)

\(=-\left(x^4-4x^2+4\right)-3\)

\(=-\left(x-2\right)^2-3\)

Do : \(-\left(x-2\right)^2\le0\Rightarrow-\left(x-2\right)^2-3\le-3< 0\)

Vậy \(-x^4+4x^2-7< 0\left(\forall x\right)\)

Wish you study well !!