Question

cho a>2,b>2.chứng minh ab>a+b

Answer 1

Ta có:

\(a>2\Leftrightarrow a-2>0\)

\(b>2\Leftrightarrow b-2>0\)

\(\Leftrightarrow\left(a-2\right)\left(b-2\right)>0\Leftrightarrow ab-2a-2b+4>0\)

\(\Leftrightarrow ab+4>2a+2b\Leftrightarrow ab+4>2\left(a+b\right)\)

Lại có:

\(ab>2.2=4\Rightarrow ab+ab>ab+4>2\left(a+b\right)\)

\(\Leftrightarrow2ab>2\left(a+b\right)\Leftrightarrow ab>a+b\) (đpcm)

Answer 2

Vì: \(a>2\Rightarrow a=2+m\)
\(b>2\Rightarrow b=2+n\)
\(\Rightarrow a+b=\left(2+m\right)+\left(2+n\right)\)
\(a.b=\left(2+m\right)+\left(2+n\right)\)
\(=2.\left(2+n\right)+m.\left(2+n\right)\)
\(=4+2.n+2.m+m.n\)
\(=4+m+m+n+n+m.n\)
\(=\left(4+m+n\right)+\left(m+n+m.n\right)\)\(=\left(2+m\right)+\left(2+n\right)+\left(m+n+m.n\right)>\left(2+m\right)+\left(2+n\right)\)
=> \(a.b>a+b\left(đpcm\right)\)