Question

CMR:4 số a,b,x,y có

\(\left(a^2+b^2\right)\left(x^2+y^2\right)\ge\left(ax+by\right)^2\)

Answer 1

Ta có:

\(\left(ay-bx\right)^2\ge0\)

\(\Rightarrow a^2y^2+b^2x^2-2aybx\ge0\)

\(\Rightarrow a^2y^2+b^2x^2\ge2aybx\)

\(\Rightarrow a^2y^2+b^2x^2+a^2x^2+b^2y^2\ge2aybx+a^2x^2+b^2y^2\)

\(\Rightarrow\left(a^2x^2+a^2y^2\right)+\left(b^2y^2+b^2x^2\right)\ge\left(ax+by\right)^2\)

\(\Rightarrow a^2\left(x^2+y^2\right)+b^2\left(y^2+x^2\right)\ge\left(ax+by\right)^2\)

\(\Rightarrow\left(x^2+y^2\right)\left(a^2+b^2\right)\ge\left(ax+by\right)^2\)

\(\rightarrowđpcm\)

Answer 2

Lên gg gõ: bđt bunhiacopxki nhé bạn. Chứng minh theo cách đưa về bp.