Question

Please help me locate all absolute maxima and minima of the following functions:
f(x,y)=(x-2)^2+(y+1)^2

Answer 1

I can see that the minima occurs at (2,-1), but how do I prove this mathematically?

Answer 2

fx(x,y)=2(x-2)
2(x-2)=0
x=2

Answer 3

fy(x,y)=2(y+1)
2(y+1)=0
y=-1

Answer 4

Is that right? how do I prove it's the minimum?

Answer 5

It is right, indeed. Stationary points occur when the first derivative is equal to 0.

Answer 6

fx(x,y)=fy(x,y)=0

Answer 7

Ok and how do I know it is a min and not a max?

Answer 8

I know it is a min by looking at the formula but how do I prove this?

Answer 9

The second test derivative. Remember for single variable calculus? It is a maximum if f'(a)=0 and f''(a)<0. For minimum, f''(a)>0.

Answer 10

woops. The second derivative test, I meant.

Answer 11

what if the second derivative has a variable in it? do I plug the x value I found when fx=0 and then plug that into the second derivative?

Answer 12

It's a bit more complicated than that... But in this case you won't have a variable on it, as fxx is equal to 2, and fyy is 2 as well.

Answer 13

You can have saddle points, which are stationary points but are not minimum nor maximum points.

Answer 14

One second... I have something that might be of your interest.

Answer 15

http://i.imgur.com/ze2jc1m.png

Answer 16

This is the general guideline to check for stationary points.

Answer 17

ok thanks a lot

Answer 18

No problem. I'm studying Calculus II as well :P