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对于一些采用常规方法难以证明的不等式,若通过适当的变形构造出典型的函数,然后利用导数研究其单调性,进而再利用其单调性即可快速获证,这种方法不仅别开生面,富有新意,而且对于提高同学们思维的灵活性、敏捷性与创造性也非常有益.例1设f_n(x)=x~n-(x-a)~n,对任意正整数n≥a>0,证明:f′_(n+1)(n+1)>(n+1)f′_n(n).分析虽然本题所涉及的数学知识不是太多,但采用常规方法很难奏效,很多同学会感到无所适从、无处入手.若采用导数法予以证明,则很容易、轻松.
For some inequalities that are difficult to prove by conventional methods, if a typical function is constructed through appropriate deformation and then its derivative is used to study its monotonicity and then its monotony can be quickly obtained, this method is not only unique and innovative, But also is very useful for improving the flexibility, agility and creativity of students ’thinking.Example 1 Let f_n (x) = x ~ n- (xa) ~ n, for any positive integer n≥a> 0, prove that: f’ (n + 1) (n + 1)> (n + 1) f’_n (n). Analysis Although the mathematical knowledge involved in this study is not too much, it is very difficult to use the conventional method. Many students feel confused, Nowhere to start.If you use the derivative method to prove it is very easy and easy.