help for extremum points

Hi,

I have this function p:R^N₊ -> [0,1]. And I am trying to show that it has a local maximum at x*=(x_1,x_2_,..x_N) such that x_1=x_2=...x_N. I already proved that no other vector satisfies the N first order conditions so the only family of extremum points is x*. Moreover p(x*) = A, where 1>A>0 is a scalar for all such x*.

The problem I face is that the Hessian matrix at x* is negative semi-definite, thus I cannot conclude whether it is a maximum or a saddle point. In my search I came across the ''Extremum test'' in Wolfram site however it applies only to x \in R; not for N variables.

I plotted the function and it obviously reaches a maximum there for any parameter involved, however I cannot prove that it is not a saddle point. (Headbang)

I would appreciate if anyone could tell me a reference or a possible solution out of this.

Hey ayildizp.

What information do you have about other turning points? Can you use the implicit function theorem (and the Jacobian) to show that if you have finite "turning points" where the Jacobian is zero, that this particular one is a global maximum?