Finding Local/Absolute Extrema of f(x,y)=x^2+y^2

[chunkyman343]

]I have to find the local/absolute extrema for the following function:
f(x,y)=x^2+y^2

bounded by the triangle x=0,y=0,y+2x=2

So far i have:
fx(x,y)=2x
fy(x,y)=2y
fxx(x,y)=2
fyy(x,y)=2
fxy&fyx(x,y)=0

critical pts at (0,0,0)
domain: 0<=x<=1, 0<=y<=2

i don't know what i should do next?

 

[Office_Shredder]

Since it's only a function of two dimensions, the critical point you get is actually at (0,0). Notice that there are precisely two points where a maximum or a minimum can occur: At a critical point, or on the boundary of the domain. So now you have to check the boundary for any maxima (that you found a critical point on the boundary is pure coincidence; the rest of the boundary still needs to be checked)

 

[chunkyman343]

So this is what i get:

local max: (0,2,4)
abs. min:(1,0,1)
abs. max:(1,2,5)

 

[Office_Shredder]

What happened to the critical point you found earlier?

 

[chunkyman343]

(0,0,0) is an absolute min.

how does that look now?

 

[chunkyman343]

with (1,0,1) being a local min.