x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.
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![x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.](/uploads/image/z/3837578-50-8.jpg?t=x%2Cy%E6%98%AF%E6%AD%A3%E6%95%B0%2C%E6%B1%82%E8%AF%81%3A%28x%2By%29%28x%5E2%2By%5E2%29%28x%5E3%2By%5E3%29%E5%A4%A7%E4%BA%8E%E7%AD%89%E4%BA%8E8x%5E3y%5E3.)
x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.
x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.
x,y是正数,求证:(x+y)(x^2+y^2)(x^3+y^3)大于等于8x^3y^3.
x+y≥2√(xy)
x²+y²≥2√(x²y²)
x³+y³≥2√(x³y³)
三式相乘,因为都是当x=y时取等号,故得到的不等式也能取等号
(x+y)(x²+y²)(x³+y³)≥8√(x^6y^6)=8x³y³
(√X+√Y)²>0,(X+Y)²>0
直接用均值定理就行。