已知函数f(x)=-acos2x-2√3asinxcosx+2a+b的定义域为【0,2/π],值域为【-5,1】.求常数
已知函数f(x)=-acos2x-2√3asinxcosx+2a+b的定义域为【0,2/π],值域为【-5,1】.求常数a,b的值
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∵f(x)=-acos2x-√3asin2x+2a+b
=-2a(cos2x/2+√3/2*sin2x)+2a+b
=-2asin(2x+π/6)+2a+b
∵0≤x≤π/2∴π/6≤2x+π/6≤7π/6,
∴-1/2≤sin(2x+π/6)≤1,
当a＞0,f(x)max=a+2a+b=3a+b=1,
f(x)min=-2a+2a+b=b=-5,
解得a=2,b=-5,
当a=0,不合题意,舍去.
当a＜0,f(x)max=-2a+2a+b=b=1,
f(x)min=a+2a+b=3a+b=-5,
解得a=-2,b=1,
∴a=2,b=-5,或a=-2,b=1
=-2a(cos2x/2+√3/2*sin2x)+2a+b
=-2asin(2x+π/6)+2a+b
∵0≤x≤π/2∴π/6≤2x+π/6≤7π/6,
∴-1/2≤sin(2x+π/6)≤1,
当a＞0,f(x)max=a+2a+b=3a+b=1,
f(x)min=-2a+2a+b=b=-5,
解得a=2,b=-5,
当a=0,不合题意,舍去.
当a＜0,f(x)max=-2a+2a+b=b=1,
f(x)min=a+2a+b=3a+b=-5,
解得a=-2,b=1,
∴a=2,b=-5,或a=-2,b=1
1年前
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