已知x∈(0,π/2),求函数f(x)=(1+cos2x+8sin^2 x)/sin2x的最小值?请用高一下基本不等式的知识来解题,
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![已知x∈(0,π/2),求函数f(x)=(1+cos2x+8sin^2 x)/sin2x的最小值?请用高一下基本不等式的知识来解题,](/uploads/image/z/1265501-29-1.jpg?t=%E5%B7%B2%E7%9F%A5x%E2%88%88%EF%BC%880%2C%CF%80%2F2%EF%BC%89%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%3D%281%2Bcos2x%2B8sin%5E2+x%29%2Fsin2x%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%3F%E8%AF%B7%E7%94%A8%E9%AB%98%E4%B8%80%E4%B8%8B%E5%9F%BA%E6%9C%AC%E4%B8%8D%E7%AD%89%E5%BC%8F%E7%9A%84%E7%9F%A5%E8%AF%86%E6%9D%A5%E8%A7%A3%E9%A2%98%2C)
已知x∈(0,π/2),求函数f(x)=(1+cos2x+8sin^2 x)/sin2x的最小值?请用高一下基本不等式的知识来解题,
已知x∈(0,π/2),求函数f(x)=(1+cos2x+8sin^2 x)/sin2x的最小值?
请用高一下基本不等式的知识来解题,
已知x∈(0,π/2),求函数f(x)=(1+cos2x+8sin^2 x)/sin2x的最小值?请用高一下基本不等式的知识来解题,
cos2x=1-2sin^2x
2sin^2x=1-cos2x
8sin^2x=4-4cos2x
y=f(x)=(1+cos2x+8sin^2x)/sin2x
=(1+cos2x+4-4cos2x)/sin2x
=(5-3cos2x)/sin2x
=(5-3cos2x)/√[1-(cos2x)^2]
已知00
∴y>0
y*√[1-(cos2x)^2]=5-3cos2x
y^2*[1-(cos2x)^2]=(5-3cos2x)^2
(9+y^2)*(cos2x)^2-30(cos2x)+25-y^2=0
上方程未知数为(cos2x)的判别式△≥0,即
(-30)^2-4*(9+y^2)*(25-y^2)≥0
y^4-16y^2≥0
y^2*(y+4)*(y-4))≥0
y≥4(另一解y≤-4舍去)
y的最小值=4
y=4
(9+y^2)*(cos2x)^2-30(cos2x)+25-y^2=0
(5cos2x-3)^2=0
cos2x=3/5,sin2x=4/5
y=f(x)=(1+cos2x+8sin^2x)/sin2x
=(1+cos2x+4-4cos2x)/sin2x
=(5-3cos2x)/sin2x
=(5-3*3/5)/(4/5)
=4
答:当0