a^2+b^2为什么≥0.5*(a+b)^2已知a>0,b>0,a+b=1,求证:(a+1/a)(平方)(b+1/b)(平方)≥25/2 (a+1/a)^2+(b+1/b)^2 =a^2+1/a^2+2 +b^2+1/b^2+2 =(a^2+b^2) + (1/a^2+1/b^2) +4 ≥0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4 这里看不懂=0.5+ 0.5*(1/a +
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![a^2+b^2为什么≥0.5*(a+b)^2已知a>0,b>0,a+b=1,求证:(a+1/a)(平方)(b+1/b)(平方)≥25/2 (a+1/a)^2+(b+1/b)^2 =a^2+1/a^2+2 +b^2+1/b^2+2 =(a^2+b^2) + (1/a^2+1/b^2) +4 ≥0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4 这里看不懂=0.5+ 0.5*(1/a +](/uploads/image/z/8711772-60-2.jpg?t=a%5E2%2Bb%5E2%E4%B8%BA%E4%BB%80%E4%B9%88%E2%89%A50.5%2A%28a%2Bb%29%5E2%E5%B7%B2%E7%9F%A5a%3E0%2Cb%3E0%2Ca%2Bb%3D1%2C%E6%B1%82%E8%AF%81%EF%BC%9A%EF%BC%88a%2B1%2Fa%29%EF%BC%88%E5%B9%B3%E6%96%B9%EF%BC%89%28b%2B1%2Fb%29%EF%BC%88%E5%B9%B3%E6%96%B9%EF%BC%89%E2%89%A525%2F2+%28a%2B1%2Fa%29%5E2%2B%28b%2B1%2Fb%29%5E2+%3Da%5E2%2B1%2Fa%5E2%2B2+%2Bb%5E2%2B1%2Fb%5E2%2B2+%3D%28a%5E2%2Bb%5E2%29+%2B+%281%2Fa%5E2%2B1%2Fb%5E2%29+%2B4+%E2%89%A50.5%2A%28a%2Bb%29%5E2+%2B0.5%2A%281%2Fa+%2B1%2Fb%29%5E2+%2B4+%E8%BF%99%E9%87%8C%E7%9C%8B%E4%B8%8D%E6%87%82%3D0.5%2B+0.5%2A%281%2Fa+%2B)
a^2+b^2为什么≥0.5*(a+b)^2已知a>0,b>0,a+b=1,求证:(a+1/a)(平方)(b+1/b)(平方)≥25/2 (a+1/a)^2+(b+1/b)^2 =a^2+1/a^2+2 +b^2+1/b^2+2 =(a^2+b^2) + (1/a^2+1/b^2) +4 ≥0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4 这里看不懂=0.5+ 0.5*(1/a +
a^2+b^2为什么≥0.5*(a+b)^2
已知a>0,b>0,a+b=1,求证:(a+1/a)(平方)(b+1/b)(平方)≥25/2
(a+1/a)^2+(b+1/b)^2
=a^2+1/a^2+2 +b^2+1/b^2+2
=(a^2+b^2) + (1/a^2+1/b^2) +4
≥0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4 这里看不懂
=0.5+ 0.5*(1/a +1/b)^2+4
=4.5+0.5*(1/a+1/b)^2
因为ab≤0.25*(a+b)^2=0.25,
所以1/a+1/b=(a+b)/ab=1/ab ≥4;
(1/a +1/b)^2≥16
所以(a+1/a)^2+(b+1/b)^2≥4.5+0.5*16=25/2
a^2+b^2为什么≥0.5*(a+b)^2已知a>0,b>0,a+b=1,求证:(a+1/a)(平方)(b+1/b)(平方)≥25/2 (a+1/a)^2+(b+1/b)^2 =a^2+1/a^2+2 +b^2+1/b^2+2 =(a^2+b^2) + (1/a^2+1/b^2) +4 ≥0.5*(a+b)^2 +0.5*(1/a +1/b)^2 +4 这里看不懂=0.5+ 0.5*(1/a +
√[(a^2+b^2)/2]≥(a+b)/2
把这个代回去再看看,就懂了
a^2+b^2≥0.5*(a+b)^2 0.5除过去
2(a^2+b^2)≥(a+b)^2
2(a^2+b^2)≥a^2+b^2+2ab
a^2+b^2≥2ab
(a-b)^2≥0显然成立
逆推上去即可
∵a2-2ab+b2=(a-b)2≥0
∴a2+b2≥2ab 2a2+2b2≥a2+2ab+ b2
∴a2+b2≥0.5(a+b)2
因为[(a^2+b^2)/2]≥(a+b)/2