函数f(X)=sin2X+根号3cos2X
(1)将f(X)化为Asin(就是一般式的那个)(2)求出f(X)的最大值最小值(3)求出f(X)的周期(4)求f(X)的单调区间请详细的讲解一下...
(1) 将f(X)化为Asin(就是一般式的那个) (2)求出f(X)的最大值 最小值
(3)求出f(X)的周期 (4) 求f(X)的单调区间
请详细的讲解一下 展开
(3)求出f(X)的周期 (4) 求f(X)的单调区间
请详细的讲解一下 展开
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(1)f(X)=sin2X+√3cos2X=2(1/2sin2X+√3/2cos2X)=2(cosπ/3sin2x+sinπ/3cos2x)
=2sin(2x+π/3)
即f(x)=2sin(2x+π/3)
(2)∵-1≤sin(2x+π/3)≤1 ∴-2≤2sin(2x+π/3)≤2
∴函数的最大值为2,最小值为-2
(3)最小正周期为T=2π/2=π
(4)令2x+π/3=-π/2+2kπ 得x=-5π/12+kπ
-5π/12+kπ+π/2 = π/12+kπ (增加半个周期)
π/12+kπ +π/2=7π/12+kπ
∴函数的单调增区间为[-5π/12+kπ , π/12+kπ ]
单调减区间为[π/12+kπ ,7π/12+kπ ]
=2sin(2x+π/3)
即f(x)=2sin(2x+π/3)
(2)∵-1≤sin(2x+π/3)≤1 ∴-2≤2sin(2x+π/3)≤2
∴函数的最大值为2,最小值为-2
(3)最小正周期为T=2π/2=π
(4)令2x+π/3=-π/2+2kπ 得x=-5π/12+kπ
-5π/12+kπ+π/2 = π/12+kπ (增加半个周期)
π/12+kπ +π/2=7π/12+kπ
∴函数的单调增区间为[-5π/12+kπ , π/12+kπ ]
单调减区间为[π/12+kπ ,7π/12+kπ ]
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(1):f(x)=2*(1/2sin2x+根号3/2cos2x)
=2*sin(2x+π/3)
(2):sin(2x+π/3)最大值为1 最小值为-1∴f(X)最大值为2,最小值为-2
(3):T=2π/W=2π/2=π
(4):f(x)取得最大值时sin(2x+π/3)=1∴2x=kπ+π/6
f(x)去的最小值时sin(2x+π/3)=-1∴2x=kπ+7π/6
∴f(x)的单调区间为[{(k-1)π+7π/6}/2,(kπ+π/6)/2] ∪ [(Kπ+π/6)/2,(kπ+7π/6)/2]
=2*sin(2x+π/3)
(2):sin(2x+π/3)最大值为1 最小值为-1∴f(X)最大值为2,最小值为-2
(3):T=2π/W=2π/2=π
(4):f(x)取得最大值时sin(2x+π/3)=1∴2x=kπ+π/6
f(x)去的最小值时sin(2x+π/3)=-1∴2x=kπ+7π/6
∴f(x)的单调区间为[{(k-1)π+7π/6}/2,(kπ+π/6)/2] ∪ [(Kπ+π/6)/2,(kπ+7π/6)/2]
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