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October 19th, 2013, 05:59 AM   #1
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is this function concave?

is (x1^2+x2^2)^1/2 a concave function?

In the lecture today we were going through the lagrangian multiplier and the condition was that the utility function has to be concave. But when I check this function on a 3D graph the function seems convex? Am I missing something?
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October 28th, 2013, 08:57 AM   #2
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Re: is this function concave?

If I remember correctly:

Concavity requires:


Both second derivative negative and hessian determinant negative

First derivatives
x1*(x1^2+x2^2)^-1/2
x2*(x1^2+x2^2)^-1/2

Second derivatives
(x1^2+x2^2)^-1/2 - x1^2(x1^2+x2^2)^-3/2
(x1^2+x2^2)^-1/2 - x2^2(x1^2+x2^2)^-3/2

Positive assuming Assuming x1,x2>0 which is a very natural.

Then the second derivatives are positive and they thus don't qualify for a concave function.
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