Find the extreme value (s) of z = 2x1^2 - x1x2 + 4x2^2 + x1x3 + x3^2 + 2 and using the Hessian matrix check whether the extreme value (s) is / are maximum or minimum.

12 次查看（过去 30 天）
P Menon
P Menon 2018-8-22
编辑： Carlos Guerrero García 2022-11-7
Kindly help with answer to the above question at an early date. Thanks.
  4 个评论
显示 2更早的评论
Bjorn Gustavsson
Bjorn Gustavsson 2018-8-22
Is perhaps x1, x2 and x3 three independent variables? If so you have a 3-D calculus problem. Which is nothing more than a generalization of the 1-D calculus...
...in that case it would be long-term useless to provide you with answers or solutions since this surely is an introductory problem for you to learn from?
Torsten
Torsten 2018-8-22
This problem can be solved using MATLAB, but my guess is that it is meant to be solved with pencil and paper.

请先登录，再进行评论。

请先登录，再回答此问题。

回答（1 个）

Carlos Guerrero García
编辑：Carlos Guerrero García 2022-11-7
The gradient of the function z(x1,x2,x3) is (4*x1-x2+x3,-x1+8*x2,x1+2*x3) and so, the function z has the origin as its unique critical point. Also, the hessian matrix (by rows) is [4 -1 1;-1 8 0;1 0 2] and the characteristical poly is
p(L)=-L^3+14L^2-54L+54
and the alternating sign of its coefficients (-1 14 -54 54) allow us to confirm that the origin is a local minimum.
You can also confirm that the local minimum is a global minimum observing that z function can be written as
z=2+((x1-8*x2)^2+(2*x1+4*x3)^2+(3*sqrt(3)*x1)^2)/16

类别

在 Help Center 和 File Exchange 中查找有关 Calculus 的更多信息

标签

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Translated by