Proving Convexity of f(x,y) = XαYβ

[3.141592654]

Homework Statement

f(x,y) = X\alphaY\beta

(that's X to the power of alpha, Y to the power of Beta)

Is this function convex? Prove it.

Homework Equations

f''(x,y) > 0 ==>convexity

The Attempt at a Solution

My steps are as follows:

f(X,Y) = (X^a)(Y^ß)

f’(X,Y) = (aX^a-1)(Yb) + (X^a)(bY^b-1)

f’’(X,Y) = ((a^(2)-a)X^a-2)(Yb) + (aX^a-1)(bY^b-1) + (aX^a-1)(bY^b-1) + (X^a)((b^(2)-b)Y^b-2)

I'm just trying to simplify it to prove it is greater than 0 (or not). Is my work correct so far and how can I be sure the second derivative is indeed positive? Thanks a lot.

 

[tiny-tim]

f(x,y) = X\alphaY\beta

Is this function convex? Prove it.

f''(x,y) > 0 ==>convexity

Hi 3.141592654!

(have an alpha: α and a beta: β
… and look above the reply box for the X² tag )

What is your ' differentiating with respect to?

You have a 3-D surface, with two variables.

 

[daviddoria]

Is it possible to use the definition of convexity to show a function is convex? Or do you just have to show the 2nd derivative is positive for all x?

Dave