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用WolframAlpha求多元函数的极值

2011-09-02 16:12:56| 分类： WolframAlpha | 标签： |举报 |字号大中小订阅

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二元函数的极值

用WolframAlpha求多元函数的极值 - calculus - 高等数学

输入：plot x^3-y^3+3*x^2+3*y^2-9*x , x=-4..4,y=-4..4

网址：http://www.wolframalpha.com/input/?i=plot++x%5E3-y%5E3%2B3*x%5E2%2B3*y%5E2-9*x+%2C+x%3D-4..4%2Cy%3D-4..4

求函数的驻点（stationary point）：stationary point of x^3-y^3+3*x^2+3*y^2-9*x

http://www.wolframalpha.com/input/?i=stationary+point+of++x%5E3-y%5E3%2B3*x%5E2%2B3*y%5E2-9*x

以上结果告诉我们：函数有四个驻点。其中(1,0)是极小值点，(-3,2)是极大值点，

(-3,0)和（1,2）是鞍点（saddle point）（非极值点）。

求极大值：maximum of x^3-y^3+3*x^2+3*y^2-9*x 或 maximize x^3-y^3+3*x^2+3*y^2-9*x

http://www.wolframalpha.com/input/?i=maximum+of++x%5E3-y%5E3%2B3*x%5E2%2B3*y%5E2-9*x

求极小值：minimum of x^3-y^3+3*x^2+3*y^2-9*x 或 minimize x^3-y^3+3*x^2+3*y^2-9*x

http://www.wolframalpha.com/input/?i=minimize++x%5E3-y%5E3%2B3*x%5E2%2B3*y%5E2-9*x

最值

输入：minimize 2(xy+2/x+2/y) for x>0, y>0 或 global minimum of 2(xy+2/x+2/y) for x>0, y>0

http://www.wolframalpha.com/input/?i=minimize+2%28xy%2B2%2Fx%2B2%2Fy%29++for+x%3E0%2C+y%3E0

用WolframAlpha求多元函数的极值 - calculus - 高等数学

输入：global maximum of 24xsin(y)-2x^2sin(y)+x^2sin(y)cos(y) for 12>x>0, Pi/2>y>0

http://www.wolframalpha.com/input/?i=global+maximum+of+24xsin%28y%29-2x%5E2sin%28y%29%2Bx%5E2sin%28y%29cos%28y%29++for+12%3Ex%3E0%2C+Pi%2F2%3Ey%3E0

先求最大值。输入：maximize x^2+2y^2-x in x^2+y^2<=1

网址：http://www.wolframalpha.com/input/?i=maximize+x%5E2%2B2y%5E2-x+in+x%5E2%2By%5E2%3C%3D1

再求最小值。输入：minimize x^2+2y^2-x in x^2+y^2<=1

网址：http://www.wolframalpha.com/input/?i=minimize+x%5E2%2B2y%5E2-x+in+x%5E2%2By%5E2%3C%3D1

三元函数的极值

三元函数的极值的求法较难，一般本科教材中是没有讲的。
可见我的这个博客：http://xuxzmail.blog.163.com/blog/static/2513191620073128266729/

先求极小值。输入：local minima x+y-e^x-e^y+2e^z-e^(z^2)

网址：http://www.wolframalpha.com/input/?i=local+minima+x%2By-e%5Ex-e%5Ey%2B2e%5Ez-e%5E%28z%5E2%29

用WolframAlpha求多元函数的极值 - calculus - 高等数学

没有极小值。

再求极大值。输入：local maxima x+y-e^x-e^y+2e^z-e^(z^2)

网址：http://www.wolframalpha.com/input/?i=local+maxima+x%2By-e%5Ex-e%5Ey%2B2e%5Ez-e%5E%28z%5E2%29

在(0,0,1)处有极大值e-2。

最后看看是否为最大值。输入：maximize x+y-e^x-e^y+2e^z-e^(z^2)

网址：http://www.wolframalpha.com/input/?i=maximize+x%2By-e%5Ex-e%5Ey%2B2e%5Ez-e%5E%28z%5E2%29

函数没有最大值，只有极大值。

返回《用WolframAlpha做微积分》总目录：

http://xuxzmail.blog.163.com/blog/static/251319162011730102030274/

用WolframAlpha求多元函数的极值 - calculus - 高等数学

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