已知a＞0,函数f(x)=-2asin(2x+π/6)+2a+b,当x属于[0,π/2]时,-5≤f(x)≤1.求f(x)的单调区间

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已知a＞0,函数f(x)=-2asin(2x+π/6)+2a+b,当x属于[0,π/2]时,-5≤f(x)≤1.求f(x)的单调区间
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已知a＞0,函数f(x)=-2asin(2x+π/6)+2a+b,当x属于[0,π/2]时,-5≤f(x)≤1.求f(x)的单调区间
已知a＞0,函数f(x)=-2asin(2x+π/6)+2a+b,当x属于[0,π/2]时,-5≤f(x)≤1.求f(x)的单调区间

已知a＞0,函数f(x)=-2asin(2x+π/6)+2a+b,当x属于[0,π/2]时,-5≤f(x)≤1.求f(x)的单调区间
(1)因为,x∈［0,π/2］,
2x＋π/6∈［π/6,7π/6］,
sin(2x＋π/6)∈［-1/2,1］,
又 a＞0
所以,-2a＋2a＋b=-5
a＋2a＋b=1
解得：a=2,b=-5
(2) 由(1)知,f(x)＝-4sin(2x＋π/6)-1
由题意 g(x)＝f(x＋π/2)
＝-4sin(2x+π+π/6)-1
=4sin(2x＋π/6)-1＞1
即 sin(2x＋π/6)＞1/2
所以 2x＋π/6∈(2kπ+π/6,2kπ+5π/6)
单调增区间满足 2x＋π/6∈(2kπ+π/6,2kπ+π/2]
单调减区间满足 2x＋π/6∈[2kπ+π/2,2kπ+5π/6)
解得 g(x)的单调增区间为 (kπ,kπ+π/6]
单调减区间为 [kπ+π/6,kπ+π/3]

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