Multiple variables - check understanding

[Niles]

Homework Statement

A function g(x,y) = 3x^2-x-y+y^2. I have to find the minimum and maximum of the function on D = [0,1] x [0,1].

The Attempt at a Solution

First I have to check the ends; I mean (0,0), (0,1), (1,0) and (1,1). I also have to check the points, where the gradient is zero: (1/6,1/2).

I insert x = 0, x = 1, y = 0 and y = 1 in g(x,y) and differentiate and equal to zero - then I have 4 new poins to check.

I also insert x = 1/6 and y = 1/2 and do the same as above.

So totally I have 11 points to check?

 

[arildno]

Hmm, I get nine points:

1. The critical point for the open domain (0,1)*(0,1), that is (1/6,1/2)

2. The four critical points for the function evaluated at the four boundary lines x=0, y=0, x=1 and y=1

3. The four end points of the boundary lines, i.e, the corners (0,0), (0,1), (1,0), (1,1).

 

[Niles]

Ok, so I do not need to insert the critical points and evaluate them (diff. and equal zero)?

What if the gradient is something like (2x-1,1) - then only x = 0 and there's no critical points. Do I have to evaluate x = 0?

Thank you for a quick respond!