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[转载] 一个代数不等式

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mathe
发表于 2014-6-29 21:35:05 | 显示全部楼层
结论是将所有$x_1,x_2,...,x_n$各乘n后的问题
t=1对应两个数:0,2 贡献乘积-8（最多使用两次??）
t=2情况1对应三个数:0,0.4514162296451364698327947488,2.548583770354863530167205252，贡献乘积3.786781856645392948227249015(最多使用一次）
t=2情况2对应三个数:0,2.215250437021530196833871917,0.7847495629784698031661280788，贡献乘积-12.67567074553428183711613785(可以任意多次使用）
t=3对应4个数:0,2.307017519236618480036245864,2×0.5419168336818726459757413173,4×0.1522872033499090570030678703，贡献乘积-15.46378268292782988622147943(最多使用一次？也许可以两个？）
t=4对应5个数:0,2.349928651318642985553754527,1.218297519424779455160694324,0.8828682331717251216906905210,0.5489055960848524375948605796，贡献乘积-16.98091409574632203491364107(最多使用一次）
利用它们合并构成n个数即可

其中t=2的情况1可以用来改变总乘积的符号，所以最多使用1次
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2014-6-29 21:44:48 | 显示全部楼层
前面分析应该有点问题，但是最大t=4不会有问题。但是最终结论估计还是t=1的可以任意多个（因为其效率最高），而t=2的也应该有限个，而t=3的最多两个。
具体可用组合可以通过数字计算得出
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2014-6-29 22:14:40 来自手机 | 显示全部楼层
为方便起见可将t=1,2,3,4等乘积记为A2,A3,A4,A5,而t=3还有个正乘积B3,于是A5显然不如A2B3.所以t=4情况不需要。
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2014-6-29 22:16:48 来自手机 | 显示全部楼层
而n=4最大值必然为A2^2,最小值A4.n=5最大值必然A2A3最小值A2B3
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2014-6-30 07:59:47 来自手机 | 显示全部楼层
A4^2<A2^4 A4A3<B3A2^2 -A4B3<-A3A2^2 由此得出A4只能出现一次而且不会同其他模式同时使用。所以只有n=4才会用A4.
毋因群疑而阻独见　　毋任己意而废人言
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mathe
发表于 2014-6-30 08:09:31 来自手机 | 显示全部楼层
又因为B3^2<A3^2  -A3B3<-A2^3 A3^4<A2^6 所以对于n>4余下模式只有 A2^x   A2^xA3^y(2<=y<=3)  A2^3B3
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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mathe
发表于 2014-6-30 08:12:22 来自手机 | 显示全部楼层
又因为-A2^3B3<-A3^3,所以用模式B3时A2最多两个，也就是只有n=3,5,7才会用B3
毋因群疑而阻独见　　毋任己意而废人言
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mathe
发表于 2014-6-30 08:19:59 来自手机 | 显示全部楼层
所以n=6时最小A2^3最大A3^2. n =7时最小A2^2A3,最大A2^2B3.而对于n>=8,偶数时最值在为A2^(n/2),A2^(n/2-3)A3^2.而n为奇数时最值为A2^((n-3)/2)A3和A2^((n-9)/2)A3^3
毋因群疑而阻独见　　毋任己意而废人言
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cn8888
发表于 2014-6-30 08:37:30 | 显示全部楼层
http://bbs.emath.ac.cn/forum.php ... 12&fromuid=8888
7#的三角形的三个顶点也是要验算的,这样更准确一些!

可以尝试把三角形区域转化成球面区域.
同样是拉格朗日乘子法,转化后只需要一次拉格朗日乘子法就可以了,
但是计算结果多了很多,不过精确解的计算还是可能的.
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*把三维空间中的三角形转化成球面a^2+b^2+c^2==1*)
  3. fun=(2*a^2-b^2)(2*b^2-c^2)(2*c^2-a^2)+x*(a^2+b^2+c^2-1)
  4. fa=D[fun,a]
  5. fb=D[fun,b]
  6. fc=D[fun,c]
  7. fx=D[fun,x]
  8. sol1=Solve[{fa==0,fb==0,fc==0,fx==0},{a,b,c,x}]
  9. Print["显示三角形内部的最值"]
  10. sol11=fun/.sol1
  11. Print["数值化"]
  12. N@sol1
  13. N@sol11
复制代码

运算结果有些长,但是也算把问题解决了!
Out[12]= (2 a^2 - b^2) (2 b^2 - c^2) (-a^2 + 2 c^2) + (-1 + a^2 +
    b^2 + c^2) x

Out[13]= -2 a (2 a^2 - b^2) (2 b^2 - c^2) +
4 a (2 b^2 - c^2) (-a^2 + 2 c^2) + 2 a x

Out[14]= 4 b (2 a^2 - b^2) (-a^2 + 2 c^2) -
2 b (2 b^2 - c^2) (-a^2 + 2 c^2) + 2 b x

Out[15]= 4 (2 a^2 - b^2) c (2 b^2 - c^2) -
2 (2 a^2 - b^2) c (-a^2 + 2 c^2) + 2 c x

Out[16]= -1 + a^2 + b^2 + c^2

Out[17]= {{x -> -(1/9), b -> -(1/Sqrt[3]), c -> -(1/Sqrt[3]),
  a -> -(1/Sqrt[3])}, {x -> -(1/9), b -> -(1/Sqrt[3]),
  c -> -(1/Sqrt[3]), a -> 1/Sqrt[3]}, {x -> -(1/9), b -> -(1/Sqrt[3]),
   c -> 1/Sqrt[3], a -> -(1/Sqrt[3])}, {x -> -(1/9),
  b -> -(1/Sqrt[3]), c -> 1/Sqrt[3], a -> 1/Sqrt[3]}, {x -> -(1/9),
  b -> 1/Sqrt[3], c -> -(1/Sqrt[3]), a -> -(1/Sqrt[3])}, {x -> -(1/9),
   b -> 1/Sqrt[3], c -> -(1/Sqrt[3]), a -> 1/Sqrt[3]}, {x -> -(1/9),
  b -> 1/Sqrt[3], c -> 1/Sqrt[3], a -> -(1/Sqrt[3])}, {x -> -(1/9),
  b -> 1/Sqrt[3], c -> 1/Sqrt[3], a -> 1/Sqrt[3]}, {x -> 0, a -> -1,
  c -> 0, b -> 0}, {x -> 0, a -> 1, c -> 0, b -> 0}, {x -> 0, b -> -1,
   c -> 0, a -> 0}, {x -> 0, b -> 1, c -> 0, a -> 0}, {x -> 0,
  b -> -Sqrt[(2/7)], c -> -(2/Sqrt[7]), a -> -(1/Sqrt[7])}, {x -> 0,
  b -> -Sqrt[(2/7)], c -> -(2/Sqrt[7]), a -> 1/Sqrt[7]}, {x -> 0,
  b -> -Sqrt[(2/7)], c -> 2/Sqrt[7], a -> -(1/Sqrt[7])}, {x -> 0,
  b -> -Sqrt[(2/7)], c -> 2/Sqrt[7], a -> 1/Sqrt[7]}, {x -> 0,
  b -> Sqrt[2/7], c -> -(2/Sqrt[7]), a -> -(1/Sqrt[7])}, {x -> 0,
  b -> Sqrt[2/7], c -> -(2/Sqrt[7]), a -> 1/Sqrt[7]}, {x -> 0,
  b -> Sqrt[2/7], c -> 2/Sqrt[7], a -> -(1/Sqrt[7])}, {x -> 0,
  b -> Sqrt[2/7], c -> 2/Sqrt[7], a -> 1/Sqrt[7]}, {x -> 0,
  b -> -(2/Sqrt[7]), c -> -(1/Sqrt[7]), a -> -Sqrt[(2/7)]}, {x -> 0,
  b -> -(2/Sqrt[7]), c -> -(1/Sqrt[7]), a -> Sqrt[2/7]}, {x -> 0,
  b -> -(2/Sqrt[7]), c -> 1/Sqrt[7], a -> -Sqrt[(2/7)]}, {x -> 0,
  b -> -(2/Sqrt[7]), c -> 1/Sqrt[7], a -> Sqrt[2/7]}, {x -> 0,
  b -> -(1/Sqrt[7]), c -> -Sqrt[(2/7)], a -> -(2/Sqrt[7])}, {x -> 0,
  b -> -(1/Sqrt[7]), c -> -Sqrt[(2/7)], a -> 2/Sqrt[7]}, {x -> 0,
  b -> -(1/Sqrt[7]), c -> Sqrt[2/7], a -> -(2/Sqrt[7])}, {x -> 0,
  b -> -(1/Sqrt[7]), c -> Sqrt[2/7], a -> 2/Sqrt[7]}, {x -> 0,
  b -> 1/Sqrt[7], c -> -Sqrt[(2/7)], a -> -(2/Sqrt[7])}, {x -> 0,
  b -> 1/Sqrt[7], c -> -Sqrt[(2/7)], a -> 2/Sqrt[7]}, {x -> 0,
  b -> 1/Sqrt[7], c -> Sqrt[2/7], a -> -(2/Sqrt[7])}, {x -> 0,
  b -> 1/Sqrt[7], c -> Sqrt[2/7], a -> 2/Sqrt[7]}, {x -> 0,
  b -> 2/Sqrt[7], c -> -(1/Sqrt[7]), a -> -Sqrt[(2/7)]}, {x -> 0,
  b -> 2/Sqrt[7], c -> -(1/Sqrt[7]), a -> Sqrt[2/7]}, {x -> 0,
  b -> 2/Sqrt[7], c -> 1/Sqrt[7], a -> -Sqrt[(2/7)]}, {x -> 0,
  b -> 2/Sqrt[7], c -> 1/Sqrt[7], a -> Sqrt[2/7]}, {x -> 0, c -> -1,
  a -> 0, b -> 0}, {x -> 0, c -> 1, a -> 0,
  b -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), a -> -(1/3) Sqrt[5 + Sqrt[7]],
   c -> -(1/3) Sqrt[4 - Sqrt[7]],
  b -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), a -> -(1/3) Sqrt[5 + Sqrt[7]],
   c -> Sqrt[4 - Sqrt[7]]/3, b -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]),
  a -> Sqrt[5 + Sqrt[7]]/3, c -> -(1/3) Sqrt[4 - Sqrt[7]],
  b -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), a -> Sqrt[5 + Sqrt[7]]/3,
  c -> Sqrt[4 - Sqrt[7]]/3, b -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]),
  b -> -(1/3) Sqrt[4 - Sqrt[7]], c -> -Sqrt[5/9 + Sqrt[7]/9],
  a -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), b -> -(1/3) Sqrt[4 - Sqrt[7]],
   c -> Sqrt[5/9 + Sqrt[7]/9], a -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]),
  b -> Sqrt[4 - Sqrt[7]]/3, c -> -Sqrt[5/9 + Sqrt[7]/9],
  a -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), b -> Sqrt[4 - Sqrt[7]]/3,
  c -> Sqrt[5/9 + Sqrt[7]/9], a -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]),
  b -> -(1/3) Sqrt[5 + Sqrt[7]], a -> -(1/3) Sqrt[4 - Sqrt[7]],
  c -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), b -> -(1/3) Sqrt[5 + Sqrt[7]],
   a -> Sqrt[4 - Sqrt[7]]/3, c -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]),
  b -> Sqrt[5 + Sqrt[7]]/3, a -> -(1/3) Sqrt[4 - Sqrt[7]],
  c -> 0}, {x -> 4/81 (10 - 7 Sqrt[7]), b -> Sqrt[5 + Sqrt[7]]/3,
  a -> Sqrt[4 - Sqrt[7]]/3, c -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  a -> -(1/3) Sqrt[5 - Sqrt[7]], c -> -Sqrt[4/9 + Sqrt[7]/9],
  b -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]), a -> -(1/3) Sqrt[5 - Sqrt[7]],
   c -> Sqrt[4/9 + Sqrt[7]/9], b -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  a -> Sqrt[5 - Sqrt[7]]/3, c -> -Sqrt[4/9 + Sqrt[7]/9],
  b -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]), a -> Sqrt[5 - Sqrt[7]]/3,
  c -> Sqrt[4/9 + Sqrt[7]/9], b -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  b -> -(1/3) Sqrt[5 - Sqrt[7]], a -> -Sqrt[4/9 + Sqrt[7]/9], c -> 0},
  {x -> 4/81 (10 + 7 Sqrt[7]), b -> -(1/3) Sqrt[5 - Sqrt[7]],
  a -> Sqrt[4/9 + Sqrt[7]/9], c -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  b -> Sqrt[5 - Sqrt[7]]/3, a -> -Sqrt[4/9 + Sqrt[7]/9],
  c -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]), b -> Sqrt[5 - Sqrt[7]]/3,
  a -> Sqrt[4/9 + Sqrt[7]/9], c -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  b -> -(1/3) Sqrt[4 + Sqrt[7]], c -> -(1/3) Sqrt[5 - Sqrt[7]],
  a -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]), b -> -(1/3) Sqrt[4 + Sqrt[7]],
   c -> Sqrt[5 - Sqrt[7]]/3, a -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]),
  b -> Sqrt[4 + Sqrt[7]]/3, c -> -(1/3) Sqrt[5 - Sqrt[7]],
  a -> 0}, {x -> 4/81 (10 + 7 Sqrt[7]), b -> Sqrt[4 + Sqrt[7]]/3,
  c -> Sqrt[5 - Sqrt[7]]/3, a -> 0}}

During evaluation of In[12]:= 显示三角形内部的最值

Out[19]= {1/27, 1/27, 1/27, 1/27, 1/27, 1/27, 1/27, 1/27, 0, 0, 0, 0, \
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \
0, 0, 0, 2/
   81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
4/81 (10 - 7 Sqrt[7]) (-(4/9) + Sqrt[7]/9 + 1/9 (4 - Sqrt[7])) +
  2/9 (5/9 + Sqrt[7]/9) (-4 + Sqrt[7]) (-(5/9) - Sqrt[7]/9 +
     2/9 (4 - Sqrt[7])),
4/81 (10 - 7 Sqrt[7]) (-(4/9) + Sqrt[7]/9 + 1/9 (4 - Sqrt[7])) +
  2/9 (5/9 + Sqrt[7]/9) (-4 + Sqrt[7]) (-(5/9) - Sqrt[7]/9 +
     2/9 (4 - Sqrt[7])),
4/81 (10 - 7 Sqrt[7]) (-(4/9) + Sqrt[7]/9 + 1/9 (4 - Sqrt[7])) +
  2/9 (5/9 + Sqrt[7]/9) (-4 + Sqrt[7]) (-(5/9) - Sqrt[7]/9 +
     2/9 (4 - Sqrt[7])),
4/81 (10 - 7 Sqrt[7]) (-(4/9) + Sqrt[7]/9 + 1/9 (4 - Sqrt[7])) +
  2/9 (5/9 + Sqrt[7]/9) (-4 + Sqrt[7]) (-(5/9) - Sqrt[7]/9 +
     2/9 (4 - Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
2/81 (-4 + Sqrt[7]) (5 + Sqrt[7]) (1/9 (-5 - Sqrt[7]) +
     2/9 (4 - Sqrt[7])) +
  4/81 (10 - 7 Sqrt[7]) (-1 + 1/9 (4 - Sqrt[7]) + 1/9 (5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-(5/9) + Sqrt[7]/9 + 1/9 (5 - Sqrt[7])) +
  2/9 (5 - Sqrt[7]) (-(4/9) - Sqrt[7]/9) (2 (4/9 + Sqrt[7]/9) +
     1/9 (-5 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-1 + 1/9 (5 - Sqrt[7]) + 1/9 (4 + Sqrt[7])) +
  2/81 (-4 - Sqrt[7]) (5 - Sqrt[7]) (1/9 (-5 + Sqrt[7]) +
     2/9 (4 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-1 + 1/9 (5 - Sqrt[7]) + 1/9 (4 + Sqrt[7])) +
  2/81 (-4 - Sqrt[7]) (5 - Sqrt[7]) (1/9 (-5 + Sqrt[7]) +
     2/9 (4 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-1 + 1/9 (5 - Sqrt[7]) + 1/9 (4 + Sqrt[7])) +
  2/81 (-4 - Sqrt[7]) (5 - Sqrt[7]) (1/9 (-5 + Sqrt[7]) +
     2/9 (4 + Sqrt[7])),
4/81 (10 + 7 Sqrt[7]) (-1 + 1/9 (5 - Sqrt[7]) + 1/9 (4 + Sqrt[7])) +
  2/81 (-4 - Sqrt[7]) (5 - Sqrt[7]) (1/9 (-5 + Sqrt[7]) +
     2/9 (4 + Sqrt[7]))}

During evaluation of In[12]:= 数值化

Out[21]= {{x -> -0.111111, b -> -0.57735, c -> -0.57735,
  a -> -0.57735}, {x -> -0.111111, b -> -0.57735, c -> -0.57735,
  a -> 0.57735}, {x -> -0.111111, b -> -0.57735, c -> 0.57735,
  a -> -0.57735}, {x -> -0.111111, b -> -0.57735, c -> 0.57735,
  a -> 0.57735}, {x -> -0.111111, b -> 0.57735, c -> -0.57735,
  a -> -0.57735}, {x -> -0.111111, b -> 0.57735, c -> -0.57735,
  a -> 0.57735}, {x -> -0.111111, b -> 0.57735, c -> 0.57735,
  a -> -0.57735}, {x -> -0.111111, b -> 0.57735, c -> 0.57735,
  a -> 0.57735}, {x -> 0., a -> -1., c -> 0., b -> 0.}, {x -> 0.,
  a -> 1., c -> 0., b -> 0.}, {x -> 0., b -> -1., c -> 0.,
  a -> 0.}, {x -> 0., b -> 1., c -> 0., a -> 0.}, {x -> 0.,
  b -> -0.534522, c -> -0.755929, a -> -0.377964}, {x -> 0.,
  b -> -0.534522, c -> -0.755929, a -> 0.377964}, {x -> 0.,
  b -> -0.534522, c -> 0.755929, a -> -0.377964}, {x -> 0.,
  b -> -0.534522, c -> 0.755929, a -> 0.377964}, {x -> 0.,
  b -> 0.534522, c -> -0.755929, a -> -0.377964}, {x -> 0.,
  b -> 0.534522, c -> -0.755929, a -> 0.377964}, {x -> 0.,
  b -> 0.534522, c -> 0.755929, a -> -0.377964}, {x -> 0.,
  b -> 0.534522, c -> 0.755929, a -> 0.377964}, {x -> 0.,
  b -> -0.755929, c -> -0.377964, a -> -0.534522}, {x -> 0.,
  b -> -0.755929, c -> -0.377964, a -> 0.534522}, {x -> 0.,
  b -> -0.755929, c -> 0.377964, a -> -0.534522}, {x -> 0.,
  b -> -0.755929, c -> 0.377964, a -> 0.534522}, {x -> 0.,
  b -> -0.377964, c -> -0.534522, a -> -0.755929}, {x -> 0.,
  b -> -0.377964, c -> -0.534522, a -> 0.755929}, {x -> 0.,
  b -> -0.377964, c -> 0.534522, a -> -0.755929}, {x -> 0.,
  b -> -0.377964, c -> 0.534522, a -> 0.755929}, {x -> 0.,
  b -> 0.377964, c -> -0.534522, a -> -0.755929}, {x -> 0.,
  b -> 0.377964, c -> -0.534522, a -> 0.755929}, {x -> 0.,
  b -> 0.377964, c -> 0.534522, a -> -0.755929}, {x -> 0.,
  b -> 0.377964, c -> 0.534522, a -> 0.755929}, {x -> 0.,
  b -> 0.755929, c -> -0.377964, a -> -0.534522}, {x -> 0.,
  b -> 0.755929, c -> -0.377964, a -> 0.534522}, {x -> 0.,
  b -> 0.755929, c -> 0.377964, a -> -0.534522}, {x -> 0.,
  b -> 0.755929, c -> 0.377964, a -> 0.534522}, {x -> 0., c -> -1.,
  a -> 0., b -> 0.}, {x -> 0., c -> 1., a -> 0.,
  b -> 0.}, {x -> -0.420754, a -> -0.921698, c -> -0.387907,
  b -> 0.}, {x -> -0.420754, a -> -0.921698, c -> 0.387907,
  b -> 0.}, {x -> -0.420754, a -> 0.921698, c -> -0.387907,
  b -> 0.}, {x -> -0.420754, a -> 0.921698, c -> 0.387907,
  b -> 0.}, {x -> -0.420754, b -> -0.387907, c -> -0.921698,
  a -> 0.}, {x -> -0.420754, b -> -0.387907, c -> 0.921698,
  a -> 0.}, {x -> -0.420754, b -> 0.387907, c -> -0.921698,
  a -> 0.}, {x -> -0.420754, b -> 0.387907, c -> 0.921698,
  a -> 0.}, {x -> -0.420754, b -> -0.921698, a -> -0.387907,
  c -> 0.}, {x -> -0.420754, b -> -0.921698, a -> 0.387907,
  c -> 0.}, {x -> -0.420754, b -> 0.921698, a -> -0.387907,
  c -> 0.}, {x -> -0.420754, b -> 0.921698, a -> 0.387907,
  c -> 0.}, {x -> 1.40841, a -> -0.511452, c -> -0.859312,
  b -> 0.}, {x -> 1.40841, a -> -0.511452, c -> 0.859312,
  b -> 0.}, {x -> 1.40841, a -> 0.511452, c -> -0.859312,
  b -> 0.}, {x -> 1.40841, a -> 0.511452, c -> 0.859312,
  b -> 0.}, {x -> 1.40841, b -> -0.511452, a -> -0.859312,
  c -> 0.}, {x -> 1.40841, b -> -0.511452, a -> 0.859312,
  c -> 0.}, {x -> 1.40841, b -> 0.511452, a -> -0.859312,
  c -> 0.}, {x -> 1.40841, b -> 0.511452, a -> 0.859312,
  c -> 0.}, {x -> 1.40841, b -> -0.859312, c -> -0.511452,
  a -> 0.}, {x -> 1.40841, b -> -0.859312, c -> 0.511452,
  a -> 0.}, {x -> 1.40841, b -> 0.859312, c -> -0.511452,
  a -> 0.}, {x -> 1.40841, b -> 0.859312, c -> 0.511452, a -> 0.}}

Out[22]= {0.037037, 0.037037, 0.037037, 0.037037, 0.037037, 0.037037, \
0.037037, 0.037037, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \
0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \
0., 0.140251, 0.140251, 0.140251, 0.140251, 0.140251, 0.140251, \
0.140251, 0.140251, 0.140251, 0.140251, 0.140251, 0.140251, \
-0.469469, -0.469469, -0.469469, -0.469469, -0.469469, -0.469469, \
-0.469469, -0.469469, -0.469469, -0.469469, -0.469469, -0.469469}

点评

sunwukong
条件和目标函数都是关于a^2、b^2、c^2轮换对称的，所以可加入条件0<=a<=b<=c。求解sol1的语句变成sol1 = Solve[{fa == 0, fb == 0, fc == 0, fx == 0, 0 <= a <= b <= c}, {a, b, c, x}]，这样结果   发表于 2014-7-3 07:37

评分

参与人数 1威望 +2 金币 +2 贡献 +2 经验 +2 鲜花 +2 收起 理由
kastin + 2 + 2 + 2 + 2 + 2 不错

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cn8888
发表于 2014-6-30 08:45:47 | 显示全部楼层
  1. Clear["Global`*"];(*Clear all variables*)
  2. (*把三维空间中的三角形转化成球面a^2+b^2+c^2+d^2==1*)
  3. fun=(2*a^2-b^2)(2*b^2-c^2)(2*c^2-d^2)(2*d^2-a^2)+x*(a^2+b^2+c^2+d^2-1)
  4. fa=D[fun,a]
  5. fb=D[fun,b]
  6. fc=D[fun,c]
  7. fd=D[fun,d]
  8. fx=D[fun,x]
  9. sol1=Solve[{fa==0.0,fb==0.0,fc==0.0,fd==0.0,fx==0.0},{a,b,c,d,x}]
  10. Print["显示三角形内部的最值"]
  11. sol11=fun/.sol1
复制代码


四维的mathematica直接干不动了,也许是因为变量的次数太高了.
算了很长时间没有结果.

点评

kastin
应该只能数值方法了。  发表于 2014-6-30 11:27
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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