Question

如何证明当r=R时,(u/(r+R))的平方乘以R的值最大

Answer 1

[U/(r+R)]^2*R=[U√R/(r+R)]^2=[U/(r/√R+√R)]^2

由均值不等式r/√R+√R≥2√r

故[U/(r+R)]^2*R=[U/(r/√R+√R)]^2≤[U/2√r]^2=U^2/(4r)

由不等号成立条件r/√R=√R

即r=R时

[U/(r+R)]^2*R取最大值U^2/(4r)