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解函数极值问题的方法很多,本文意在通过极值的几何解释使变量之间的关系变得较为直观,从而使问题的解决较为具体、简捷。例1 求函数y=(x~2+4x+5)~(1/2)+(x~2-6x+25)~(1/2)的最小值。解式中变量y是x的无理函数,若将其有理化,计算是相当繁杂的。而要直接求解又难以下手。考查这个函数表达式的几何意义,发现可表示成两点间距离之和,即 y=((x+2)~2+1~2)~(1/2)+((x-3)~2+4~2)~(1/2)。则y表示x轴上的动点P(x,0)到两定点
There are many ways to solve the extremum problem of the function. This paper intends to make the relationship between the variables more intuitive through the geometric explanation of the extremum, so that the solution of the problem is more concrete and simple. Example 1 Find the minimum value of the function y=(x~2+4x+5)~(1/2)+(x~2-6x+25)~(1/2). The variable y in the solution is an irrational function of x. If it is rationalized, the calculation is quite complicated. It is difficult to start with a direct solution. Examine the geometric meaning of this function expression and find that it can be expressed as the sum of the distances between two points, ie y=((x+2)~2+1~2)~(1/2)+((x-3)~ 2+4~2)~(1/2). Then y represents the moving point P(x,0) on the x axis to two fixed points