题根证明:对于a、b、c、d∈R,恒有不等式(ac+bd)~2≤(a~2+b~2)(c~2+d~2).1这是人教A版“普通高中新课程标准实验教科书《数学》(必修4)”中第108页的一道习题.本文以此题为素材,开展数学研究性学习:探究此题的多种来源、多种证法、多种变式与广泛应用等.1探究多种来源此题除了上述教材来源(称为教材来源1)之外,还有下列人教A版的多种来源.教材来源2:《选修4-5》不等式选讲中第23页的 The root of the proof: For a, b, c, d∈R, constant inequality (ac+bd)~2≤(a~2+b~2)(c~2+d~2).1 A version of the “experimental textbook for mathematics” (Compulsory 4) in the New Curriculum Standards for Ordinary High Schools (“Required 4”), “An Exercise on page 108.” This article is based on the subject and conducts research on mathematics. It explores multiple sources of this problem. A variety of methods, multiple variants, and wide applications.1 Exploring Multiple Sources In addition to the sources of the above textbooks (referred to as textbook source 1), there are various sources for teaching the following versions of the A version. Selected 4-5 inequality in the selection of the 23rd page