作业帮已知a大于0,b大于0,a+b=1,求证(a+1/a)(b+1/b)大于或等于25/4.解法里面有一步不懂.(a+1/a)(b+1/b)=(a^2+1)/a*(b^2+1)/b=(a^2b^2+a^2+1+b^2)/ab=[a^2b^2+(a+b)^2-2ab+1]/ab=[a^2b^2+(1-2ab)+1]/ab=[(ab-1)^2+1]/ab(ab-1)^2+1≥25/16 #0
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/10 22:17:11
![已知a大于0,b大于0,a+b=1,求证(a+1/a)(b+1/b)大于或等于25/4.解法里面有一步不懂.(a+1/a)(b+1/b)=(a^2+1)/a*(b^2+1)/b=(a^2b^2+a^2+1+b^2)/ab=[a^2b^2+(a+b)^2-2ab+1]/ab=[a^2b^2+(1-2ab)+1]/ab=[(ab-1)^2+1]/ab(ab-1)^2+1≥25/16 #0](/uploads/image/z/1606556-20-6.jpg?t=%E5%B7%B2%E7%9F%A5a%E5%A4%A7%E4%BA%8E0%2Cb%E5%A4%A7%E4%BA%8E0%2Ca%2Bb%3D1%2C%E6%B1%82%E8%AF%81%28a%2B1%2Fa%29%28b%2B1%2Fb%29%E5%A4%A7%E4%BA%8E%E6%88%96%E7%AD%89%E4%BA%8E25%2F4.%E8%A7%A3%E6%B3%95%E9%87%8C%E9%9D%A2%E6%9C%89%E4%B8%80%E6%AD%A5%E4%B8%8D%E6%87%82.%28a%2B1%2Fa%29%28b%2B1%2Fb%29%3D%28a%5E2%2B1%29%2Fa%2A%28b%5E2%2B1%29%2Fb%3D%28a%5E2b%5E2%2Ba%5E2%2B1%2Bb%5E2%29%2Fab%3D%5Ba%5E2b%5E2%2B%28a%2Bb%29%5E2-2ab%2B1%5D%2Fab%3D%5Ba%5E2b%5E2%2B%281-2ab%29%2B1%5D%2Fab%3D%5B%28ab-1%29%5E2%2B1%5D%2Fab%28ab-1%29%5E2%2B1%E2%89%A525%2F16+%230)
已知a大于0,b大于0,a+b=1,求证(a+1/a)(b+1/b)大于或等于25/4.解法里面有一步不懂.(a+1/a)(b+1/b)=(a^2+1)/a*(b^2+1)/b=(a^2b^2+a^2+1+b^2)/ab=[a^2b^2+(a+b)^2-2ab+1]/ab=[a^2b^2+(1-2ab)+1]/ab=[(ab-1)^2+1]/ab(ab-1)^2+1≥25/16 #0
已知a大于0,b大于0,a+b=1,求证(a+1/a)(b+1/b)大于或等于25/4.解法里面有一步不懂.
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=(a^2b^2+a^2+1+b^2)/ab
=[a^2b^2+(a+b)^2-2ab+1]/ab
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16 #
0
已知a大于0,b大于0,a+b=1,求证(a+1/a)(b+1/b)大于或等于25/4.解法里面有一步不懂.(a+1/a)(b+1/b)=(a^2+1)/a*(b^2+1)/b=(a^2b^2+a^2+1+b^2)/ab=[a^2b^2+(a+b)^2-2ab+1]/ab=[a^2b^2+(1-2ab)+1]/ab=[(ab-1)^2+1]/ab(ab-1)^2+1≥25/16 #0
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=(a^2b^2+a^2+1+b^2)/ab
=[a^2b^2+(a+b)^2-2ab+1]/ab
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
∵(a+b)/2≥√(ab)
∴ab≤[(a+b)/2]²=1/4
0
ab>=(a+b)的平方/4=1/4
(ab-1)^2+1≥25/16