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高中《代数》第二册有这样一道习题:求证:(a~2+b~2)(c~2+d~2)≥(ac+bd)~2当且仅当ad=bc时取等号。它是柯西不等式的特例。当然此不等式的证明相当简单;可利用该结论解决一些不等式问题,巧妙简捷,颇具新意,常能使问题化难为易,化繁为简。例1 设a,b均为正数且a+b=1,求证:((2a+1)~(1/2))+((2b+1)~(1/2))≤2 2~(1/2)。
The second volume of “Algebra” in high school has such an exercise: Proof: (a~2+b~2)(c~2+d~2)≥(ac+bd)~2 if and only if ad=bc number. It is a special case of Cauchy inequality. Of course, the proof of this inequality is quite simple; this conclusion can be used to solve some problems of inequality, clever and simple, quite innovative, and often can make the problem difficult, easy to simplify. Example 1 Let a and b be positive numbers and a+b=1. Prove that ((2a+1)~(1/2))+((2b+1)~(1/2))≤2 2~ (1/2).