设m>0,n>0,实数a,b,c,d满足a+b+c+d=m,ac=bd=n²,求√(a+b)(b+c)(c+d)(d+a)的值(用m,n表示)1
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/19 14:06:51
![设m>0,n>0,实数a,b,c,d满足a+b+c+d=m,ac=bd=n²,求√(a+b)(b+c)(c+d)(d+a)的值(用m,n表示)1](/uploads/image/z/6974504-8-4.jpg?t=%E8%AE%BEm%3E0%2Cn%3E0%2C%E5%AE%9E%E6%95%B0a%2Cb%2Cc%2Cd%E6%BB%A1%E8%B6%B3a%2Bb%2Bc%2Bd%3Dm%2Cac%3Dbd%3Dn%26sup2%3B%2C%E6%B1%82%E2%88%9A%28a%2Bb%29%28b%2Bc%29%28c%2Bd%29%28d%2Ba%29%E7%9A%84%E5%80%BC%EF%BC%88%E7%94%A8m%2Cn%E8%A1%A8%E7%A4%BA%EF%BC%891)
xSnP(MJm_݇S!.E"ejPBB%1(/0vhsfΝ9wf\
j ɢOt1~X$"߿"PXZ~! Ԋp9`x
O^E|IGLgh>*U-ZRZ{DLxЉߟ0Fޏ7|]M)\4noDhzxfcY.k7; I44?NhܻNv([+b!YfE) &?J\V
(4֚/3qVƗ܊~PeZ¼]0+KvA{vdWn
设m>0,n>0,实数a,b,c,d满足a+b+c+d=m,ac=bd=n²,求√(a+b)(b+c)(c+d)(d+a)的值(用m,n表示)1
设m>0,n>0,实数a,b,c,d满足a+b+c+d=m,ac=bd=n²,求√(a+b)(b+c)(c+d)(d+a)的值(用m,n表示)
1
设m>0,n>0,实数a,b,c,d满足a+b+c+d=m,ac=bd=n²,求√(a+b)(b+c)(c+d)(d+a)的值(用m,n表示)1
如图,答案是mn
又因为a^2+b^2=m^2,c^2+d^2=n^2 (m>0,n>0),所以 (mn)^2>=(ac+bd)^2 开方可得(因为m,n为正数) mn>=|ac+bd| 又由基本不等式
√(a+b)(b+c)(c+d)(d+a)
=√(ab+ac+b²+bc)(cd+ac+d²+ad)
=√[ac+b(a+b+c)][ac+d(a+c+d)]
=√[n²+b(m-d)][n²+d(m-b)]
=√(n²+bm-bd)(n²+dm-bd)
=√(n²+bm-n²)(n²+dm-n²)
=√(bm)(dm)
=√(m²*n²)
=mn