Proving concavity of a 2 variable function
I was wondering how to prove f(x,y)=lnx + lny + 2x^(1/2)y^(1/2) is concave over set S (where X = (x,y):x>0). I got the 2nd derivative matrix but cannot seem to prove negative semidefitness (the 1st principal leading minors are negative...but no clue how to show the 2nd principal minor would be positive). I feel like I have to use the 1st characterization of concavity (i.e. f(Lx + (1-L)y)> Lf(x) + (1-L)f(y) where L is Lambda 0<L<1 and for every xEs and yEs, but am not sure how to use this characterization. Any help?