Lời giải:
Áp dụng BĐT Bunhiacopxky:
\(\left(\frac{x^4}{a}+\frac{y^4}{b}\right)(a+b)\geq (x^2+y^2)^2=1\)
\(\Rightarrow \frac{x^4}{a}+\frac{y^4}{b}\geq \frac{1}{a+b}\)
Dấu "=" xảy ra khi \(\frac{x^2}{a}=\frac{y^2}{b}\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\frac{x^2}{a}=\frac{y^2}{b}=\frac{x^2+y^2}{a+b}=\frac{1}{a+b}\)
\(\Rightarrow \frac{x^{2000}}{a^{1000}}+\frac{y^{2000}}{b^{1000}}=\left(\frac{x^2}{a}\right)^{1000}+\left(\frac{y^2}{b}\right)^{1000}\)
\(=\frac{1}{(a+b)^{1000}}+\frac{1}{(a+b)^{1000}}=\frac{2}{(a+b)^{1000}}\)
Chu y dua ve bieu thuc dong bac de bien doi nhe
\(\dfrac{x^4}{a}+\dfrac{y^4}{b}=\dfrac{\left(x^2+y^2\right)^2}{a+b}\)\(\Leftrightarrow\)\(\dfrac{x^4}{a}+\dfrac{y^4}{b}=\dfrac{x^4+y^4-2x^2y^2}{a+b}\)
\(\Leftrightarrow\dfrac{bx^4\left(a+b\right)+\left(a+b\right)ay^4-ab\left(x^4+y^4-2x^2y^2\right)}{ab\left(a+b\right)}=0\)
\(\Leftrightarrow\dfrac{a^2y^4+b^2x^4-2abx^2y^2}{ab\left(a+b\right)}=0\)\(\Leftrightarrow\left(ay^2-bx^2\right)^2=0\)
\(\Leftrightarrow ay^2=bx^2\Leftrightarrow\dfrac{x^2}{a}=\dfrac{y^2}{b}=\dfrac{x^2+y^2}{a+b}=\dfrac{1}{a+b}\)
\(\Leftrightarrow\dfrac{x^{2000}}{a^{1000}}=\dfrac{y^{2000}}{b^{1000}}=\dfrac{1}{\left(a+b\right)^{1000}}\)
-->QED