Question

Cho x,y,z > 0 và xyz=1 . Tìm MinP = \(\Sigma\dfrac{1}{x^4\left(y+1\right)\left(z+1\right)}\)

Answer 1

Đặt \(\left(x;y;z\right)=\left(\dfrac{1}{a};\dfrac{1}{b};\dfrac{1}{c}\right)\Rightarrow abc=1\)

\(P=\sum\dfrac{a^4}{\left(\dfrac{1}{b}+1\right)\left(\dfrac{1}{c}+1\right)}=\sum\dfrac{a^4bc}{\left(b+1\right)\left(c+1\right)}=\sum\dfrac{a^3}{\left(b+1\right)\left(c+1\right)}\)

Ta có:

\(\dfrac{a^3}{\left(b+1\right)\left(c+1\right)}+\dfrac{b+1}{8}+\dfrac{c+1}{8}\ge\dfrac{3a}{4}\)

Tương tự và cộng lại:

\(P+\dfrac{a+b+c}{4}+\dfrac{3}{4}\ge\dfrac{3\left(a+b+c\right)}{4}\Rightarrow P\ge\dfrac{a+b+c}{2}-\dfrac{3}{4}\ge\dfrac{3}{2}-\dfrac{3}{4}=\dfrac{3}{4}\)