函数f(x)=2asin²x-2根号3asinxcosx+a+b,x∈[0,π/2],值域为[-5,1],求a、b
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![函数f(x)=2asin²x-2根号3asinxcosx+a+b,x∈[0,π/2],值域为[-5,1],求a、b](/uploads/image/z/2535279-15-9.jpg?t=%E5%87%BD%E6%95%B0f%EF%BC%88x%EF%BC%89%3D2asin%26sup2%3Bx-2%E6%A0%B9%E5%8F%B73asinxcosx%2Ba%2Bb%2Cx%E2%88%88%5B0%2C%CF%80%2F2%5D%2C%E5%80%BC%E5%9F%9F%E4%B8%BA%5B-5%2C1%5D%2C%E6%B1%82a%E3%80%81b)
函数f(x)=2asin²x-2根号3asinxcosx+a+b,x∈[0,π/2],值域为[-5,1],求a、b
函数f(x)=2asin²x-2根号3asinxcosx+a+b,x∈[0,π/2],值域为[-5,1],求a、b
函数f(x)=2asin²x-2根号3asinxcosx+a+b,x∈[0,π/2],值域为[-5,1],求a、b
2sin^2x=1-cos2x
2sinxcosx=sin2x
f(x)=a-acos2x-a√3sin2x+a+b
=2a+b-2a(cos2xcosπ/3+sin2xsinπ/3)
=2a+b-2acos(2x-π/3)
-5
f(x)=a(1-cos2x)-sqrt(3)asin2x+a+b
=-2a[(1/2)(cos2x)+(sqrt(3)/2)sin2x]+2a+b
=-2asin(2x+π/6)+2a+b
x∈[0,π/2] => sin(2x+π/6)∈[-1/2,1];
显然a不等于0;
当a>0时:f(x)∈[-2a+2a+b,a+2a+b]=[b,...
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f(x)=a(1-cos2x)-sqrt(3)asin2x+a+b
=-2a[(1/2)(cos2x)+(sqrt(3)/2)sin2x]+2a+b
=-2asin(2x+π/6)+2a+b
x∈[0,π/2] => sin(2x+π/6)∈[-1/2,1];
显然a不等于0;
当a>0时:f(x)∈[-2a+2a+b,a+2a+b]=[b,3a+b]=[-5,1]
=> a=2,b=1;
当a<0时:f(x)∈[a+2a+b,-2a+2a+b]=[3a+b,b]=[-5,1]
=> a=-2,b=1;
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