a, Chứng minh với mọi a.b ta có : 2(a2 + b2 ) \(\ge\)( a + b )2
b, x,y,x dương thỏa mãn \(\sqrt{x}\)+ \(\sqrt{y}\)+ \(\sqrt{z}\)= 1 . Chứng minh rằng :
\(\sqrt{\frac{xy}{x+y+2z}}\)+ \(\sqrt{\frac{yz}{y+z+2x}}\)+ \(\sqrt{\frac{zx}{z+x+2y}}\)\(\le\)\(\frac{1}{2}\)
Dấu "=" xảy ra khi nào?
\(a,\)\(2\left(a^2+b^2\right)\ge\left(a+b\right)^2\)
\(\Rightarrow2a^2+2b^2\ge a^2+2ab+b^2\)
\(\Rightarrow a^2+b^2\ge2ab\)
\(\Rightarrow a^2-2ab+b^2\ge0\)
\(\Rightarrow\left(a-b\right)^2\ge0\) ( luôn đúng )
\(\Rightarrow2\left(a^2+b^2\right)\ge\left(a+b\right)^2\)