一道求极值问题，真不会

xfhaoym

求(2xy+3)/(xy^2)+(y^2+xy)/(x+3y)的极值（不管大小）有什么办法吗？

nyy

求导数解方程组，下周看看

mathe

好像是五次方程

笨笨

实数根式解



小数解

nyy

1. Clear["Global`*"];(*清除所有变量*)
   
2. f=(2*x*y+3)/(x*y^2)+(y^2+x*y)/(x+3*y)
   
3. fx=D[f,x]
   
4. fy=D[f,y]
   
5. ans=Solve[{fx,fy}==0,{x,y},Reals]
   
6. aaa=Append[#,FullSimplify[f/.#]]&/@ans
   
7. (f/.ans)//FullSimplify(*目标函数值*)
   
8. Grid[N@aaa,Alignment->Left](*列表显示*)
   

复制代码

可能的四个极值
\[\left\{\frac{1}{9} (-4) \sqrt{20 \sqrt{\frac{2}{3}}-2 \sqrt{6 \sqrt{6}+\frac{38}{3}}+18},\frac{4}{9} \sqrt{20 \sqrt{\frac{2}{3}}-2 \sqrt{6 \sqrt{6}+\frac{38}{3}}+18},\frac{4}{9} \sqrt{\frac{2}{3} \left(10 \sqrt{6}+\sqrt{6 \left(9 \sqrt{6}+19\right)}+27\right)},\frac{1}{9} (-4) \sqrt{\frac{2}{3} \left(10 \sqrt{6}+\sqrt{6 \left(9 \sqrt{6}+19\right)}+27\right)}\right\}\]

矩阵数值,第一列x，第二列y，第三列函数值
\[\begin{array}{lll}
x\to -1.32415 & y\to -3.1621 & -2.17132 \\
x\to 1.32415 & y\to 3.1621 & 2.17132 \\
x\to -10.7376 & y\to 0.948736 & 2.97452 \\
x\to 10.7376 & y\to -0.948736 & -2.97452 \\
\end{array}\]

代码再改进改进

1. Clear["Global`*"];(*清除所有变量*)
   
2. f=(2*x*y+3)/(x*y^2)+(y^2+x*y)/(x+3*y)
   
3. fx=D[f,x]
   
4. fy=D[f,y]
   
5. ans=Solve[{fx,fy}==0,{x,y},Reals]//FullSimplify
   
6. aaa=Append[#,FullSimplify[f/.#]]&/@ans
   
7. Grid[aaa,Alignment->Left](*列表显示*)
   
8. Grid[N@aaa,Alignment->Left](*列表显示*)
   

复制代码

第一列x，第二列y，第三列函数值
\[\begin{array}{lll}
x\to -\sqrt{\frac{3}{5} \left(19 \sqrt{6}-\sqrt{30 \left(61 \sqrt{6}+149\right)}+51\right)} & y\to -\sqrt{\sqrt{6}+\sqrt{6 \left(\sqrt{6}+1\right)}+3} & \frac{1}{9} (-4) \sqrt{20 \sqrt{\frac{2}{3}}-2 \sqrt{6 \sqrt{6}+\frac{38}{3}}+18} \\
x\to \sqrt{\frac{3}{5} \left(19 \sqrt{6}-\sqrt{30 \left(61 \sqrt{6}+149\right)}+51\right)} & y\to \sqrt{\sqrt{6}+\sqrt{6 \left(\sqrt{6}+1\right)}+3} & \frac{4}{27} \sqrt{60 \sqrt{6}-6 \sqrt{6 \left(9 \sqrt{6}+19\right)}+162} \\
x\to -\sqrt{\frac{3}{5} \left(19 \sqrt{6}+\sqrt{30 \left(61 \sqrt{6}+149\right)}+51\right)} & y\to \sqrt{\sqrt{6}-\sqrt{6 \left(\sqrt{6}+1\right)}+3} & \frac{4}{9} \sqrt{\frac{2}{3} \left(10 \sqrt{6}+\sqrt{6 \left(9 \sqrt{6}+19\right)}+27\right)} \\
x\to \sqrt{\frac{3}{5} \left(19 \sqrt{6}+\sqrt{30 \left(61 \sqrt{6}+149\right)}+51\right)} & y\to -\sqrt{\sqrt{6}-\sqrt{6 \left(\sqrt{6}+1\right)}+3} & \frac{1}{9} (-4) \sqrt{\frac{2}{3} \left(10 \sqrt{6}+\sqrt{6 \left(9 \sqrt{6}+19\right)}+27\right)} \\
\end{array}\]

数值值
\[\begin{array}{lll}
x\to -1.3241533520525920905 & y\to -3.1621003372960161574 & -2.1713230852996915144 \\
x\to 1.3241533520525920905 & y\to 3.1621003372960161574 & 2.1713230852996915144 \\
x\to -10.737550187808406456 & y\to 0.94873649789537297262 & 2.9745248662862202539 \\
x\to 10.737550187808406456 & y\to -0.94873649789537297262 & -2.9745248662862202539 \\
\end{array}\]

nyy

你缺少的不是数学知识，缺乏的是软件

xfhaoym

nyy 发表于 2023-8-4 10:10
你缺少的不是数学知识，缺乏的是软件

谢谢，我就不会软件。