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设f(x)=(sin^4 x-cos^4 x-5)/(cos2x+2). (a)证明f(x)= 3/(2 sin^2 x-3) -1 (b)求f(x)的范围
人气:436 ℃ 时间:2020-04-07 05:48:14
解答
f(x)=(sin^4 x-cos^4 x-5)/(cos2x+2)
=(sin^2 x-cos^2 x-5)/(cos2x+2)
=-(cos2x+5)/(cos2x+2)
=-1-3/(cos2x+2)
=-1-3/(1-2sin^2 x+2)
=3/(2sin^2x-3)-1
证毕
b)、f(x)==3/(2sin^2x-3)-1,sin^2x∈【0,1】
所以2sin^2x-3∈【-3,-1】
所以f(x)max=-2,f(x)min=-4
即f(x)的值域为【-4,-2】
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