Question

Chứng minh 

a) x² -2x+3>0 với mọi x 

b) x² + 7x + 13 >0 với mọi x 

c) x-x² -1 <0 với mọi x 

Answer 1

a, x²-2x+3

=x²-2x+1+2

=(x-1)²+2

\(\Rightarrow\left(x-1\right)^2\ge0\)voi moi x

Dpcm

b, x²+7x+13 

=x²+7x+\(\frac{49}{4}\)+\(\frac{3}{4}\)

=\(\left(x+\frac{7}{2}\right)^2+\frac{3}{4}\)

\(\Rightarrow\left(x+\frac{7}{2}\right)^2\ge0\)voi moi x

Dpcm

c, x-x²-1

=-x²+x-1

=\(-x^2+2.\frac{1}{2}x-\frac{1}{4}+\frac{5}{4}\)

=\(-\left(x-\frac{1}{2}\right)^2+\frac{5}{4}\)

\(=\frac{5}{4}-\left(x-\frac{1}{2}\right)^2\)

\(\Rightarrow-\left(x-\frac{1}{2}\right)^2\le0\)

Dpcm

nho k nha