已知函数f(x)=2sin^2x+2倍根号3sinxcosx+a(a属于R) (1)若x属于R,求f(x)的单调递增区间及对称中心(2)若x属于[0,π/2]时,f(x)的最小值为3,求a的值(3)若a=-1,由函数y=f(x)的图像经过平移能否得到一个
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![已知函数f(x)=2sin^2x+2倍根号3sinxcosx+a(a属于R) (1)若x属于R,求f(x)的单调递增区间及对称中心(2)若x属于[0,π/2]时,f(x)的最小值为3,求a的值(3)若a=-1,由函数y=f(x)的图像经过平移能否得到一个](/uploads/image/z/4002128-8-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2sin%5E2x%2B2%E5%80%8D%E6%A0%B9%E5%8F%B73sinxcosx%2Ba%28a%E5%B1%9E%E4%BA%8ER%29+%EF%BC%881%EF%BC%89%E8%8B%A5x%E5%B1%9E%E4%BA%8ER%2C%E6%B1%82f%28x%29%E7%9A%84%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E5%8C%BA%E9%97%B4%E5%8F%8A%E5%AF%B9%E7%A7%B0%E4%B8%AD%E5%BF%83%EF%BC%882%EF%BC%89%E8%8B%A5x%E5%B1%9E%E4%BA%8E%5B0%2C%CF%80%2F2%5D%E6%97%B6%2Cf%28x%29%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%E4%B8%BA3%2C%E6%B1%82a%E7%9A%84%E5%80%BC%EF%BC%883%EF%BC%89%E8%8B%A5a%3D-1%2C%E7%94%B1%E5%87%BD%E6%95%B0y%3Df%28x%29%E7%9A%84%E5%9B%BE%E5%83%8F%E7%BB%8F%E8%BF%87%E5%B9%B3%E7%A7%BB%E8%83%BD%E5%90%A6%E5%BE%97%E5%88%B0%E4%B8%80%E4%B8%AA)
已知函数f(x)=2sin^2x+2倍根号3sinxcosx+a(a属于R) (1)若x属于R,求f(x)的单调递增区间及对称中心(2)若x属于[0,π/2]时,f(x)的最小值为3,求a的值(3)若a=-1,由函数y=f(x)的图像经过平移能否得到一个
已知函数f(x)=2sin^2x+2倍根号3sinxcosx+a(a属于R)
(1)若x属于R,求f(x)的单调递增区间及对称中心
(2)若x属于[0,π/2]时,f(x)的最小值为3,求a的值
(3)若a=-1,由函数y=f(x)的图像经过平移能否得到一个奇函数的图像?若能,请写出平移过程并求出最小平移距离;若不能,请说明理由.
已知函数f(x)=2sin^2x+2倍根号3sinxcosx+a(a属于R) (1)若x属于R,求f(x)的单调递增区间及对称中心(2)若x属于[0,π/2]时,f(x)的最小值为3,求a的值(3)若a=-1,由函数y=f(x)的图像经过平移能否得到一个
f(x) = 2sin^2x+2根号3sinxcosx+a = 1-cos2x+根号3sin2x+a
= 2(根号3/2 sin2x - 1/2 cos2x) + a-1 = 2( sin2xcosπ/6 - cos2xsinπ/6) + a-1
= 2sin(2x-π/6) + a-1 = 2sin2(x-π/12) + a-1
(1)
当2kπ-π/2<2(x-π/12)<2kπ+π/2,即kπ-π/6<x<kπ+π/3,k∈Z时,单调增
对称中心:(π/12,a-1)
(2)
x∈[0,π/2]时,在(0,π/3)单调增,在(π/3,π/2)单调减
f(0)=2sin(0-π/6) + a-1 =-1+a-1=a-2
f(π/2)=2sin(π-π/6) + a-1 =1+a-1=a
f(0)<f(π/2)
最小值f(0)=a-2=3,a=5
(3)
若a=-1
f(x)=2sin2(x-π/12) -1-1 = 2sin2(x-π/12) -2
将函数图像左移π/12个单位,上移2个单位,将得到一个奇函数:
f(x)= 2sin2x