Reference request for Newton's method on a convex/concave function

Question

The proof featured at http://planetmath.org/newtonsmethodworksforconvexrealfunctions shows that, roughly speaking, Newton's method is unconditionally convergent for convex/concave functions.

I am looking for a textbook or paper reference of this fact. I have found part of it in Michael Spivak's Calculus:

Answer 1

I have not found any references from books, but the cached version of a PlanetMath page (currently down) captures all of the relevant cases (attached). I am posting it to answer the question for other readers.