a) Áp dụng bđt \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\) (1)
\(\frac{1}{y}+\frac{1}{z}\ge\frac{4}{y+z}\) (2)
Cộng vế vs vế (1);(2) ta có:
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}\ge\frac{4}{x+y}+\frac{4}{y+z}\)
Mà: \(\frac{4}{x+y}+\frac{4}{y+z}=4\left(\frac{1}{x+y}+\frac{1}{y+z}\right)\ge4\left(\frac{4}{x+2y+z}\right)=\frac{16}{x+2y+z}\)
=> \(\frac{16}{x+2y+z}\le\frac{1}{x}+\frac{2}{y}+\frac{1}{z}\)
=> \(\frac{1}{x+2y+z}\le\frac{1}{16}\left(\frac{1}{x}+\frac{2}{y}+\frac{1}{z}\right)\)
=> \(\frac{y}{x+2y+z}\le\frac{y}{16}\left(\frac{1}{x}+\frac{2}{y}+\frac{1}{z}\right)\) (3)
Tương tự ta cũng có:
\(\frac{x}{2x+y+z}\le\frac{x}{16}\left(\frac{2}{x}+\frac{1}{y}+\frac{1}{z}\right)\) (4)
\(\frac{z}{x+y+2z}\le\frac{z}{16}\left(\frac{1}{x}+\frac{1}{y}+\frac{2}{z}\right)\) (5)
Từ (3);(4);(5) suy ra:
\(\frac{x}{2x+y+z}+\frac{y}{x+2y+z}+\frac{z}{x+y+2z}\le\frac{1}{16}\left(2+\frac{x}{y}+\frac{x}{z}+\frac{y}{x}+2+\frac{y}{z}+\frac{z}{x}+\frac{z}{y}+2\right)\)
Vì: \(x,y,z>0\) nên áp dụng bđt cô-si ta có:
\(\frac{x}{y}+\frac{y}{x}\ge2;\frac{y}{z}+\frac{z}{y}\ge2;\frac{x}{z}+\frac{z}{x}\ge2\)
Do đó:
\(\frac{x}{2x+y+z}+\frac{y}{x+2y+z}+\frac{z}{x+y+2z}\le\frac{1}{16}\left(6+2+2+2\right)=\frac{1}{16}\cdot12=\frac{3}{4}\)
b)
Vì: \(ab+bc+ca\le\\ a^2+b^2+c^2=\left(a+b+c\right)^2-2\left(ab+bc+ca\right)=-2\left(ab+bc+ca\right)\)
=> \(3\left(ab+bc+ca\right)\le0\)
=> \(ab+bc+ca\le0\)
