Question

Multivariable max-min problem:
Find the max and min values of the function f(x,y)=x^2+y^2 when x, y belong to the line x+y=1.
So, i followed the Lagrange multipliers method and ended with this single point: (1/2,1/2). How I'm supposed to determine whether this is a max a min o none of them?.

Answer 1

There is no way to know the type of extrema using lagrange multipliers method

Answer 2

you need to find the type by other means

Answer 3

Not even with the Hessian matrix?

Answer 4

why do you want to use hessian matrix if figuring out the type of extrema is easy simply by staring at the function ?

Answer 5

Pick a nearby point on the line \(x+y=1\) and evaluate \(f(x,y)\) there. Is it larger or smaller than \(f(1/2,1/2)\)? That should tell you what you need to know, no?

Answer 6

if you want to practice hessian matrix stuff, you may do it that way...
otherwise, i feel using hessian matrix to figure out the type of extrema is a sledgehammer in present problem

Answer 7

I know is a minima, but i'm looking for a general (analitic) method for the case when the constraint functions are not closed sets. For use in more complicated problems.

Answer 8

Ahh okay then hessian matrix should work nicely

Answer 9

\[|H| = \begin{vmatrix}0&1&1\\1&2&0\\1&0&2 \end{vmatrix} = -4\]

so any critical point is a minimum

Answer 10

as a second approach , you can write y=1-x
and put it in f=x^2+(1-x)^2
then you have single variable problem
\[f=x^2+(1-x)^2=\\2x^2-2x+1\\f'=4x-2=0\\x=\frac{1}{2} \to y=\frac{1}{2}\]

Answer 11

and this is obviously min ,because f''=4>0