Question

Cho các số a,b,x,y là các số thực thỏa mãn:x²+y²=1 và\(\frac{x^4}{a}+\frac{y^4}{b}=\frac{1}{a+b}\)

Chứng minh rằng:\(\frac{x^6}{a^3}+\frac{y^6}{b^3}=\frac{2}{\left(a+b\right)^3}\)

Answer 1

Ta co:

\(\frac{x^4}{a}+\frac{y^4}{b}\ge\frac{\left(x^2+y^2\right)^2}{a+b}=\frac{1}{a+b}\)

Dau '=' xay ra khi \(\frac{x^2}{a}=\frac{y^2}{b}\)

Ta lai co:

\(\frac{x^6}{a^3}+\frac{y^6}{b^3}=\left(\frac{x^2}{a}\right)^3+\left(\frac{y^2}{b}\right)^3=2\left(\frac{x^2}{a}\right)^3\)

Ma \(\frac{x^2}{a}=\frac{y^2}{b}=\frac{x^2+y^2}{a+b}=\frac{1}{a+b}\)

\(\Rightarrow x^2=\frac{a}{a+b}\)

\(\Leftrightarrow\frac{x^2}{a}=\frac{1}{a+b}\)

\(\Leftrightarrow\left(\frac{x^2}{a}\right)^3=\frac{1}{\left(a+b\right)^3}\)

\(\Rightarrow\frac{x^6}{a^3}+\frac{y^6}{b^3}=\frac{2}{\left(a+b\right)^3}\)