试题分析:(I)将函数F(x)=f(x)f′(x)+f
2(x)化一可得:F(x)=1+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144584344.png)
sin(2x+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144631396.png)
),由此可得F(x)的最小正周期及单调区间.(Ⅱ) 由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144647745.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144662924.png)
这样可得sin(2x+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144631396.png)
)的范围,从而得函数F(x)的值域.
(Ⅲ)由f(x)=2f′(x),得:sinx+cosx=2cosx-2sinx,由此可得tanx的值.
将
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144522882.png)
化为只含tanx式子,将tanx.的值代入即可.
试题解析:(I)∵f′(x)=cosx-sinx,
∴F(x)=f(x)f′(x)+f
2(x)=cos
2x-sin
2x+1+2sinxcosx=1+sin2x+cos2x=1+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144584344.png)
sin(2x+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144631396.png)
),
最小正周期为T=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144881465.png)
=π.
单调递增区间:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144538848.png)
单调递减区间:
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144928999.png)
. 4分
(Ⅱ)由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144647745.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144662924.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144974941.png)
,所以函数F(x)的值域为[1,1+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144584344.png)
]. 8分
(Ⅲ)∵f(x)=2f′(x), ∴sinx+cosx=2cosx-2sinx,
∴cosx=3sinx, ∴tanx=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024145006327.png)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144522882.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024145037986.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024145052740.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024145068548.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824024144600362.png)
. 13分
考点:1、三角变换;2、三角函数的单调性和范围;3、三角函数同角关系式.