Question

[•๛♡长เℓℓëɾ•✰ツ]

Chứng minh : ( a^2 + b^2 ) (c^2 + d^2 ) lớn hơn hoặc bằng ( ac+bd )^2 

Answer 1

[I - Vy Nguyễn]

Ta có : \(\left(a^2+b^2\right)\left(c^2+d^2\right)\ge\left(ac+bd\right)^2\)

\(\Leftrightarrow a^2.c^2+a^2.d^2+b^2.c^2+b^2.d^2\ge\left(ac\right)^2+2acbd+\left(bd\right)^2\) 

\(\Leftrightarrow\left(ac\right)^2+\left(ad\right)^2+\left(bc\right)^2+\left(bd\right)^2\ge\left(ac\right)^2+2acbd+\left(bd\right)^2\)

\(\Leftrightarrow\left(ad\right)^2+\left(bc\right)^2\ge2acbd\)

\(\Leftrightarrow\left(ad\right)^2-2acbd+\left(bc\right)^2\ge0\)

\(\Leftrightarrow\left(ac-bd\right)^2\ge0\)

\(\Rightarrow\)luôn đúng 

Dấu " = " xảy ra khi \(\frac{a}{c}=\frac{b}{d}\)