# Concavity/Convexity of a function

• Aug 6th 2010, 01:42 PM
momomath
Concavity/Convexity of a function
I need help determining whether the following function is concave or convex:

f(x,y) = x^a * y^b ; a>0, b>0

The way to solve this is to get the determinant of the Hessian. I have narrowed it down the the result below.

[ (a-1)a(x^(a-2))y^b * (b-1)b(y^(b-2))x^a ] - [ a(x^(a-1)) * a(x^(a-1)) ]^2

I guess it is now a matter of determining whether the determinant below is >0, <0, >=0, or<=0. It is convex if the determinant is >= 0 and concave if it is <= 0. I know that when x or y = 0, the determinant is 0. I am having trouble though finding out what the result would be if I look at other combinations of of x and y values without plugging in different values and solving. If that is the only solution then I could do that, but I have a feeling there is perhaps a quicker way which I would appreciate if someone could shed some light on. Also, I hope my assumptions above are correct.

Thank you.