How to show that the Cobb-Douglas function is concave

It turned out that my attempt to explain how to show that the C-D function is concave without differentiating was not correct after all. Sorry about that!

The problem is more difficult than what I first thought, as you need to use a result related to quasiconcavity.

The argument is as follows: we have f(x)=exp(lnA + ln x1 + lnx2 + lnx3)). Now, the argument in the paranthesis is concave, and therefore also quasiconcave. Thus, Theorem 2.5.2 in the English textbook (Setning 4.7.2 in the Norwegian Version) asserts that f(x) is also quasiconcave. Then you can apply Theorem 2.5.3 (Setning 4.7.3) to conclude that f(x) must also be concave, since all the conditions of the theorem are satisfied (not very difficult to check).