【错解分析】
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240022062311049.png)
=a
2+b
2+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206278396.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206293396.png)
+4
≥2ab+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206309479.png)
+4
≥4
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206325581.png)
+4=8,
∴(a+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206340327.png)
)
2+(b+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206356344.png)
)
2的最小值是8.
上面的解答中,两次用到了基本不等式a
2+b
2≥2ab,第一次等号成立的条件是a=b=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206371338.png)
,第二次等号成立的条件是ab=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206403418.png)
,显然,这两个条件是不能同时成立的。因此,8不是最小值。
【正解】原式= a
2+b
2+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206278396.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206293396.png)
+4
="(" a
2+b
2)+(
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206278396.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206293396.png)
)+4
=[(a+b)
2-2ab]+[(
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206340327.png)
+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206356344.png)
)
2-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206309479.png)
]+4
= (1-2ab)(1+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206543494.png)
)+4,
由ab≤(
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206559531.png)
)
2=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206574303.png)
得:1-2ab≥1-
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206371338.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206371338.png)
, 且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206543494.png)
≥16,1+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206543494.png)
≥17,
∴原式≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206371338.png)
×17+4=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206683424.png)
(当且仅当a=b=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206371338.png)
时,等号成立),
∴(a +
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206340327.png)
)
2 + (b +
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206356344.png)
)
2的最小值是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824002206247422.png)
。
【点评】在应用重要不等式求解最值时,要注意它的三个前提条件缺一不可即“一正、二定、三相等”,在解题中容易忽略验证取提最值时的使等号成立的变量的值是否在其定义域限制范围内。