câu 1: a/b+c + b/c+a + c/a+b >= 1.5 câu 2: cho x>=0; y >= 0; z >=0 cmr (x+y).(y+z).(x+z)>=8xyz

Đáp án:

Giải thích các bước giải:

câu 1:

$\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\\

=\dfrac{a^{2}}{ab+ac}+\dfrac{b^{2}}{bc+ba}+\dfrac{c^{2}}{ca+cb}\\

=(\dfrac{a^{2}}{ab+ac}+\dfrac{b^{2}}{bc+ba}+\dfrac{c^{2}}{ca+cb})((ab+ac)+(bc+ba)+(ca+cb))\\

\geq (\sqrt[]{\frac{a^{2}}{ab+ac}.(ab+ac)}+\sqrt[]{\frac{b^{2}}{bc+ba}.(bc+ba)}+\sqrt[]{\frac{c^{2}}{ca+cb}.(ca+cb)})^{2}\\

=(a+b+c)^{2}\geq 3(ab+bc+ca)\\

\rightarrow \dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\geq \dfrac{3(ab+bc+ca)}{(ab+ac)+(bc+ba)+(ca+cb)}=\dfrac{3}{2}$

Câu 2:

$x+y\geq 2\sqrt[]{xy}\\

y+z\geq 2\sqrt[]{yz}\\

z+x\geq 2\sqrt[]{xz}\\

\rightarrow (x+y)(y+z)(z+x)\geq 2\sqrt[]{xy}.2\sqrt[]{yz}.2\sqrt[]{xz}=8xyz$