Question

Cho a < b < c < d

CMR : (a + b)(c + d) < (a + c)(b + d)

Answer 1

Điều cần chứng minh:\(\left(a+b\right)\left(c+d\right)< \left(a+c\right)\left(b+d\right)\)

\(\Leftrightarrow ac+ad+bc+bd< ab+ad+bc+cd\)

\(\Leftrightarrow ac+bd< ab+cd\)

\(\Leftrightarrow ac+bd-ab-cd< 0\)

\(\Leftrightarrow a\left(c-b\right)-d\left(c-b\right)< 0\)

\(\Leftrightarrow\left(c-b\right)\left(a-d\right)< 0\)

Vì a < b < c < d nên: \(\left\{{}\begin{matrix}c-b>0\\a-d< 0\end{matrix}\right.\Rightarrow\left(c-b\right)\left(a-d\right)< 0\)

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