试题分析:
(1)该函数显然是二次函数,开口向上,所以在对称轴左侧单调递减,右侧单调递增.根据题意可知区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758244493.png)
在对称轴的左侧,所以根据对称轴即可求出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758260283.png)
的取值范围;
(2)由于该二次函数的对称轴未知,所以当对称轴与区间处于不同位置时,函数的单调性会发生改变,从而影响到函数的最值,所以得讨论区间与对称轴的位置关系,通过讨论位置关系确定单调性和最值,建立关于
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758447423.png)
的关系式,从而得到最终的结论.
试题解析:
(1)该函数显然是二次函数,开口向上,所以在对称轴左侧单调递减,
该函数的对称轴为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758463505.png)
,所以区间
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758244493.png)
在对称轴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758463505.png)
的左侧,
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758541511.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758354424.png)
(2)显然
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758588398.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240547586031029.png)
,对称轴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758619502.png)
讨论对称轴与区间的位置关系:
(1)当对称轴在区间左侧时,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758634519.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758650403.png)
,此时函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758213463.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758681436.png)
上单调递增,
所以要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758712682.png)
恒成立,只需满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240547587281254.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758744429.png)
及
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758759400.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758775731.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758790590.png)
矛盾,舍.
(2)当对称轴在区间右侧时,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758806554.png)
,此时函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758213463.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758681436.png)
上单调递减,
要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758712682.png)
恒成立,只需满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240547587281254.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758900540.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758915613.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240547589311250.png)
与
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758946598.png)
矛盾,舍.
(3)当对称轴在区间内时,有
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758962631.png)
,此时函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758213463.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758993527.png)
上递减,在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759009560.png)
上递增,
要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758712682.png)
恒成立,只需满足
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240547590402330.png)
由前二式得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759056489.png)
,由后二式得
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758962631.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759102707.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759134692.png)
,故
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759165478.png)
。当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759180372.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054759196395.png)
时满足题意.
综上
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758322285.png)
的最大值为3,此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824054758385399.png)