点P(x,y)在椭圆x^2+2y^2=3上运动
∴x^2+2y^2=3即x^2=3-2y^2
则1/x^2+2/(1+y^2)
=1/(3-2y^2)+2/(1+y^2)
=1/5〔1/(3-2y^2)+4/(2+2y^2)(3-2y^2+2+2y^2)
=1/5〔5+(2+2y^2)/(3-2y^2)+4(3-2y^2)/(2+2y^2)〕》1/5〔5+2√4)=9/5
∴1/x^2+2/1+y^2的最小值为9/5.
点P(x,y)在椭圆x^2+2y^2=3上运动
∴x^2+2y^2=3即x^2=3-2y^2
则1/x^2+2/(1+y^2)
=1/(3-2y^2)+2/(1+y^2)
=1/5〔1/(3-2y^2)+4/(2+2y^2)(3-2y^2+2+2y^2)
=1/5〔5+(2+2y^2)/(3-2y^2)+4(3-2y^2)/(2+2y^2)〕》1/5〔5+2√4)=9/5
∴1/x^2+2/1+y^2的最小值为9/5.