设向量a=(sinx,cosx)b=(sinx,根号3sinx),x属于R,函数f(x)=a.(a
设向量a=(sinx,cosx)b=(sinx,根号3sinx),x属于R,函数f(x)=a.(a+2b)。求函数f(x)的最大值与单调递增区间。...
设向量a=(sinx,cosx)b=(sinx,根号3sinx),x属于R,函数f(x)=a.(a+2b)。求函数f(x)的最大值与单调递增区间。
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f(x)
= a.(a+2b)
=(sinx,cosx). (3sinx, cosx +2√3sinx)
=3(sinx)^2+ (cosx)^2 +2√3sinx.cosx
=(3/2)( 1- cos2x) + (1/2)(1+cos2x) + √3sin2x
=2[ (√3/2)sin2x - (1/2)cos2x ] + 3/2
=2sin(2x-π/6) + 3/2
max f(x) = 2 + 3/2 = 7/2
单调递增区间
2kπ - π/2 ≤2x-π/6≤ 2kπ + π/2
kπ - π/6 ≤x≤ kπ + 4π/3
= a.(a+2b)
=(sinx,cosx). (3sinx, cosx +2√3sinx)
=3(sinx)^2+ (cosx)^2 +2√3sinx.cosx
=(3/2)( 1- cos2x) + (1/2)(1+cos2x) + √3sin2x
=2[ (√3/2)sin2x - (1/2)cos2x ] + 3/2
=2sin(2x-π/6) + 3/2
max f(x) = 2 + 3/2 = 7/2
单调递增区间
2kπ - π/2 ≤2x-π/6≤ 2kπ + π/2
kπ - π/6 ≤x≤ kπ + 4π/3
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