1. ## Maximum and Minimum

Hey.
I need to find maximum and minimum for this function, as you can see, I found (0,0) as a "suspicious point" (I don't know how it's called in English ).
My question is, how can I determine if it's Max or Min (if it's possible)?
I know the test of the second derivative, but how should I get it? I mean should I use the definition of the derivative because I think there is a problem around the point (0,0).
Any idea guys?

2. Originally Posted by asi123
Hey.
I need to find maximum and minimum for this function, as you can see, I found (0,0) as a "suspicious point" (I don't know how it's called in English ).
My question is, how can I determine if it's Max or Min (if it's possible)?
I know the test of the second derivative, but how should I get it? I mean should I use the definition of the derivative because I think there is a problem around the point (0,0).
Any idea guys?
This has a maximum at $\displaystyle (0,0)$ it is a critical point as the derivative does not exist there. You need only obsever that for any $\displaystyle (x,y) \ne (0,0)$, $\displaystyle \sqrt{x^2+y^2}>0$ and so $\displaystyle z(0,0)$ must be a maximum.

($\displaystyle z(x,y)$ is smooth everywere else and there are no critical points way from $\displaystyle (0,0)$ so there are no other maxima or minima, the surface is in fact a cone with vertex at $\displaystyle (0,0)$ )

RonL