已知向量
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814769203.png)
=(
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814784219.png)
sin2
x+2,cos
x),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814800199.png)
=(1,2cos
x),設(shè)函數(shù)
f(
x)=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814769203.png)
·
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814800199.png)
.
(I)求
f(
x)的最小正周期與單調(diào)遞增區(qū)間;
(Ⅱ)在△ABC中,a,b,c分別是角A,B,C的對邊,若a=
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814784219.png)
,
f(A)=4,求b+c的最大值.
(1)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814862298.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814894226.png)
的單調(diào)遞增區(qū)間為
(2)當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814925256.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814940220.png)
最大為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814956237.png)
試題分析:解:(Ⅰ)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240128149871024.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814987365.png)
3分
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814894226.png)
的最小正周期
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814862298.png)
4分
由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815081528.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815096455.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814894226.png)
的單調(diào)遞增區(qū)間為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814909469.png)
6分
(Ⅱ)由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815143261.png)
得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815159401.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815174378.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815190249.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815206392.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815221334.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815237252.png)
8分
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815252285.png)
法一:又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815252387.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815268859.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815284434.png)
∴當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814925256.png)
時(shí),
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814940220.png)
最大為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012814956237.png)
12分
法二:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815346356.png)
即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815362956.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815377391.png)
;當(dāng)且僅當(dāng)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824012815377216.png)
時(shí)等號成立。 12分
點(diǎn)評:解決的關(guān)鍵是結(jié)合向量的數(shù)量積表示三角關(guān)系式,然后借助于三角函數(shù)的性質(zhì)來得到求解,屬于基礎(chǔ)題。
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