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.

Kindly help with answer to the above question at an early date. Thanks.

[KALYAN ACHARJYA]

Define x1,x2 & X3, Also Share What you tried so far?

[P Menon]

I have no idea about this. Kindly help with the answer / solution.

[Bjorn Gustavsson]

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]

This problem can be solved using MATLAB, but my guess is that it is meant to be solved with pencil and paper.

Answer 1

[Carlos Guerrero García]

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