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使用均值不等式求函数最值时,常常碰到不可能取等号的时候,此时,只要我们稍作变形,就能使等号成立.这就需要我们从不可能中探可能,化不可能为可能,下面举例剖析. 例1 求函数y=x2+3/(x2+2)~(1/2)的最小值。 解析y=x2+2+1/(x2+2)~(1/2)=(x2+2)~(1/2)+1/(x2+2)~(1/2)≥2,当且仅当(x2+2)~(1/2)=1/(x2+2)~(1/2)即x2+2=1(*),x2=-1时取等号,这是不可能的. 探讨 将(*)式改为x2+2=2,得x2=0,
When we use the mean inequality to find the function’s maximum value, we often find it impossible to take the equal sign. At this time, as long as we make a slight deformation, we can make the equal sign. This requires us to explore the impossible and make it impossible. For example, consider the following example. Example 1 Find the minimum value of the function y=x2+3/(x2+2)~(1/2). Parse y=x2+2+1/(x2+2)~(1/2)=(x2+2)~(1/2)+1/(x2+2)~(1/2)≥2 when And only when (x2+2)~(1/2)=1/(x2+2)~(1/2), that is, x2+2=1(*), x2=-1, the equal sign is taken, which is not Possible. Discuss changing (*) to x2+2=2, get x2=0,