I have this function p:RN+ -> [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.
I would appreciate if anyone could tell me a reference or a possible solution out of this.