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用初等数学的方法求函数的极值,陈振宜、范会国两同志分别著有小册子作了专門的叙述,其中都提到了二次函数的极值及其求法。本文旨在用同样初等的方法,对二次函数的极值問題作更深入一步的討論。有趣的是,通过极值存在与否的討論,还可以得到函数的分类;从几何的观点来看,就是曲綫的分类。陈振宜同志已經提出:如果x的函数y可以化成 f_1(y)x~2+f_2(y)x+f_3(y)=0 (1)形式,其中f_1,f_2,f_3是y的函数,那么,由于x必須是实数,即(1)必須有实根,其判別式 f_2(y)~2-4f_1(y)f_3(y)≥0。(2)只要这一不等式可解,y的极值就可以求得(見“极大与极小”第13頁)。从中学生的知識水平来看,如果(2)的左边部分是y的二次函数,那么不等式(2)总是容易解出的,而且有权現成的方法。下面三种情况都是属于这种情形
Using the method of elementary mathematics to find the extremum of the function, the two pamphlets of Chen Zhenyi and Fan Huiguo made special descriptions on pamphlets, both of which mentioned the extremum of quadratic function and its method of seeking. The purpose of this paper is to discuss the extremum of quadratic functions in more depth using the same elementary method. Interestingly, the classification of functions can also be obtained through the discussion of the existence of extreme values; from a geometric point of view, it is the classification of curves. Comrade Chen Zhenyi has proposed that if the function y of x can be converted into the form f_1(y)x~2+f_2(y)x+f_3(y)=0(1), where f_1,f_2,f_3 are functions of y, then Since x must be a real number, ie (1) must have a real root, its discriminant f_2(y)~2-4f_1(y)f_3(y)≥0. (2) As long as this inequality is solvable, the extreme value of y can be found (see “Maximum and Minimal” on page 13). From the perspective of secondary school students’ knowledge, if the left part of (2) is a quadratic function of y, then inequality (2) is always easy to solve, and it has the right to use the ready-made method. The following three situations are all belong to this situation