How Are Critical Points Classified in Calculus?

[Hjensen]

Hello,

I had a lecture in an elementary Calculus course today, and the professor used a method for classifying critical points which I can't see in my book (Calculus, a complete course, by Adams). Basically he had the function

$f(x,y)=x^2+2y^2-4x+4y.$

He found a critical point at (2,-1) - I have no problem understanding this part. However, he classified it as a local (and absolute) minimum because the function $f(x,y)$ would diverge for $x^2+y^2\to \infty$. Where does this method come from and where can I read about it?

 

[fauboca]

Hello,

I had a lecture in an elementary Calculus course today, and the professor used a method for classifying critical points which I can't see in my book (Calculus, a complete course, by Adams). Basically he had the function

$f(x,y)=x^2+2y^2-4x+4y.$

He found a critical point at (2,-1) - I have no problem understanding this part. However, he classified it as a local (and absolute) minimum because the function $f(x,y)$ would diverge for $x^2+y^2\to \infty$. Where does this method come from and where can I read about it?

Think of a parabola in 2D. If it opens up, it has an absolute min at the vertex right?

Think of a cubic. It has local or relative max and mins but at the end points it goes on to infinity so there is no absolute.

If you understand the shape of your graph, you can identify these points.

http://en.wikipedia.org/wiki/Second_partial_derivative_test