已知实数m,n满足m^2 n^2=a,x,yx满足^2 y^2=b其中a,b为常数,求mx ny最小值已知实数m,n满足m^2+n^2=a,x,y满足x^2+y^2=b其中a,b为常数,求mx+ny最小值
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![已知实数m,n满足m^2 n^2=a,x,yx满足^2 y^2=b其中a,b为常数,求mx ny最小值已知实数m,n满足m^2+n^2=a,x,y满足x^2+y^2=b其中a,b为常数,求mx+ny最小值](/uploads/image/z/2680544-56-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%AE%9E%E6%95%B0m%2Cn%E6%BB%A1%E8%B6%B3m%5E2+n%5E2%3Da%2Cx%2Cyx%E6%BB%A1%E8%B6%B3%5E2+y%5E2%3Db%E5%85%B6%E4%B8%ADa%2Cb%E4%B8%BA%E5%B8%B8%E6%95%B0%2C%E6%B1%82mx+ny%E6%9C%80%E5%B0%8F%E5%80%BC%E5%B7%B2%E7%9F%A5%E5%AE%9E%E6%95%B0m%2Cn%E6%BB%A1%E8%B6%B3m%5E2%2Bn%5E2%3Da%2Cx%2Cy%E6%BB%A1%E8%B6%B3x%5E2%2By%5E2%3Db%E5%85%B6%E4%B8%ADa%EF%BC%8Cb%E4%B8%BA%E5%B8%B8%E6%95%B0%EF%BC%8C%E6%B1%82mx%2Bny%E6%9C%80%E5%B0%8F%E5%80%BC)
已知实数m,n满足m^2 n^2=a,x,yx满足^2 y^2=b其中a,b为常数,求mx ny最小值已知实数m,n满足m^2+n^2=a,x,y满足x^2+y^2=b其中a,b为常数,求mx+ny最小值
已知实数m,n满足m^2 n^2=a,x,yx满足^2 y^2=b其中a,b为常数,求mx ny最小值
已知实数m,n满足m^2+n^2=a,x,y满足x^2+y^2=b其中a,b为常数,求mx+ny最小值
已知实数m,n满足m^2 n^2=a,x,yx满足^2 y^2=b其中a,b为常数,求mx ny最小值已知实数m,n满足m^2+n^2=a,x,y满足x^2+y^2=b其中a,b为常数,求mx+ny最小值
这种题型一般都用三角函数的方法来解答,根据题意,可设
m=√asinA,n=√acosA
x=√bsinB,y=√bcosB
所以:
mx+ny=√(ab)sinAsinB+√(ab)cosAcosB=√(ab)cos(A-B)
因此mx+ny=的最大值是√(ab),即根号下ab
这里利用了(sinA)^2+(cosA)^2=1
类似的题型很多,这种方法在以后高等数学中也有广泛应用.
如果要快的话,利用柯西不等式
(m^2+n^2)(x^2+y^2)>=(mx+ny)^2
|mx+ny|<=√(ab)
mx+ny>=-√(ab )
即最小值是-√(ab )
上式同样可以用均值不等式得到
n^2x^2+m^2y^2>=2nxmy
两边都加上m^2x^2+n^2y^2
m^2x^2+n^2y^2+n^2x^2+m^2y...
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如果要快的话,利用柯西不等式
(m^2+n^2)(x^2+y^2)>=(mx+ny)^2
|mx+ny|<=√(ab)
mx+ny>=-√(ab )
即最小值是-√(ab )
上式同样可以用均值不等式得到
n^2x^2+m^2y^2>=2nxmy
两边都加上m^2x^2+n^2y^2
m^2x^2+n^2y^2+n^2x^2+m^2y^2>=m^2x^2+n^2y^2+2nxmy
即(m^2+n^2)(x^2+y^2)>=(mx+ny)^2
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