作业帮已知函数y=(sinx)^2+根号3/2 sin2x+2(cosx)^2,求函数最小值若x属于[-π/6,π/4],求y的取值范围.
来源：学生作业帮助网 编辑：作业帮 时间：2024/05/08 21:36:42
![已知函数y=(sinx)^2+根号3/2 sin2x+2(cosx)^2,求函数最小值若x属于[-π/6,π/4],求y的取值范围.](/uploads/image/z/1043310-30-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0y%3D%28sinx%29%5E2%2B%E6%A0%B9%E5%8F%B73%2F2+sin2x%2B2%28cosx%29%5E2%2C%E6%B1%82%E5%87%BD%E6%95%B0%E6%9C%80%E5%B0%8F%E5%80%BC%E8%8B%A5x%E5%B1%9E%E4%BA%8E%5B-%CF%80%2F6%2C%CF%80%2F4%5D%2C%E6%B1%82y%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4.)
已知函数y=(sinx)^2+根号3/2 sin2x+2(cosx)^2,求函数最小值若x属于[-π/6,π/4],求y的取值范围.
已知函数y=(sinx)^2+根号3/2 sin2x+2(cosx)^2,求
函数最小值
若x属于[-π/6,π/4],求y的取值范围.
已知函数y=(sinx)^2+根号3/2 sin2x+2(cosx)^2,求函数最小值若x属于[-π/6,π/4],求y的取值范围.
y=sin²x+√3/2sin2x+2cos²x
=√3/2sin2x+cos²x+1
=√3/2sin2x+1/2(2cos²x-1)+3/2
=√3/2sin2x+1/2cos2x+3/2
=sin(2x+π/6)+3/2
最小值为：1/2
∵x∈[-π/6,π/4]
∴2x+π/6∈[-π/6,5π/6]
∴sin(2x+π/6)∈[-1/2,1]
∴y∈[1,5/2]
y=sin^2x+cos^2x+cos^2x+√3/2sin2x
=1+cos^2x+√3/2sin2x
=1+1/2·cos2x-1/2+√3/2sin2x
=1/2+sin(2x+π/6)
故ymin=1/2-1=-1/2
若x∈[-π/6,π/4]，则2x+π/6∈[-π/6,2π/3]，故y∈[0,3/2]第三部不懂，是否化错了？1/2·...
全部展开
y=sin^2x+cos^2x+cos^2x+√3/2sin2x
=1+cos^2x+√3/2sin2x
=1+1/2·cos2x-1/2+√3/2sin2x
=1/2+sin(2x+π/6)
故ymin=1/2-1=-1/2
若x∈[-π/6,π/4]，则2x+π/6∈[-π/6,2π/3]，故y∈[0,3/2]
收起
可以化成y=1+根号3/2 sin2x+cosx)^2。。。。。。。。(sinx)^2+(cosx)^2=1
=根号3/2 sin2x+（cos2x+1）/2...................cos2x=2(cosx)^2-1
=sin（2x+六分之π）+1/2 x属于[-π/6,π/4]
2x属于[...
全部展开
可以化成y=1+根号3/2 sin2x+cosx)^2。。。。。。。。(sinx)^2+(cosx)^2=1
=根号3/2 sin2x+（cos2x+1）/2...................cos2x=2(cosx)^2-1
=sin（2x+六分之π）+1/2 x属于[-π/6,π/4]
2x属于[-π/3,π/2]
2x+六分之π属于[-π/6,2π/3]
作图得出
y属于[-1/2,1]，x=-π/6时最小，x=π/2时最大
收起