实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/21 10:40:32
![实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值](/uploads/image/z/6927600-48-0.jpg?t=%E5%AE%9E%E6%95%B0x%E3%80%81y%E6%BB%A1%E8%B6%B34x%5E2-5xy%2B4y%5E2%3D5%2C%E8%AE%BES%3Dx%5E2%2By%5E2%2C%E6%B1%82S%E7%9A%84%E6%9C%80%E5%80%BC)
实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
x=√s cosB y=√s sinB
4x^2-5xy+4y^2=5
4(√s cosB)^2-5√s cosB*√s sinB+4(√s sinB)^2=5
4s (cosB)^2-5s sinBcosB+4s (sinB)^2=5
4s-5s/2 sin2B=5
因为:-1《sin2B《1
所以:s=5/(4-5/2 sin2B)∈[10/13,10/3]
所以s的最大值为:10/3,最小值为:10/13
-s/2<=xy<=s/2
-5s/2<=-5xy<=5s/2
4s-5s/2<=5,4s+5s/2>=5
3s/2<=5,13s/2>=5
10/13<=s<=10/3,