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January 已知:a,b属于正实数.求证:a/根号下b+b/根号下a>=根号下a+根号下b.
已知:a,b属于正实数.求证:a/根号下b+b/根号下a>=根号下a+根号下b.
令 x = √a、y =√b
x^3 + y^3
= (x + y)(x^2 - xy + y^2)
原式左边
= x^2/y + y^2/x
= (x^3 + y^3)/(x*y)
= (x + y)(x^2 - xy + y^2)/(xy)
≥(x + y).......x^2 - xy + y^2 ≥ 2xy - xy = xy
x + y = √a + √b
证明完毕
最佳答案 - 由投票者2008-06-05 12:43:20选出
做差
为了简单，让根号a=a，根号b=b
则:
a^2/b-b+b^2/a-a=(a^2-b^2)/b-(a^2-b^2)/a
=(a^2-b^2)(1/b-1/a)
=(a^2-b^2)(a-b)/ab
=(a-b)^2(a+b)/ab
因为a,b为正实数，所以a-b>=0，ab>0,a+b>0
所以
a^2/b-b+b^2/a-a>=0
即所求成立
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