试题分析:(1)根据题意,只要sin(x
0+1)=sinx
0+sin1成立即可,由解析式列出方程,再由特殊角的正弦值进行证明;
(2)把解析式代入f(x+1)=f(x)+f(1),列出对应的方程,再由一元二次方程有解的条件求出k的范围,注意二次系数是否为零;
(3)根据定义只要证明f(x+1)=f(x)+f(1)有解,把解析式代入列出方程,转化为对应的函数,利用函数的零点存在性判定理进行判断..
(Ⅰ)若
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738511606.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738838457.png)
,在定义域内存在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738479321.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738854917.png)
, ∵方程
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738869611.png)
无解,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738511606.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738791475.png)
。
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240007389162939.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738932388.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738947369.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738963418.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000739181427.png)
,得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240007392281411.png)
。
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738807730.png)
。
(Ⅲ)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240007392592605.png)
,
又∵函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738573479.png)
图象与函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738745408.png)
的图象有交点,设交点的横坐标为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738557283.png)
,
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000739306997.png)
,其中
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000739322492.png)
。
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738495871.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824000738760909.png)
。
点评:本题属于新定义,新情景的问题,主要利用新定义进行运算,考查了对数函数、正弦函数和指数函数的性质,函数的零点存在性判定理的应用,综合性强、难度大.