Let f:R^2 -> R be a real function which is concave and C^infinity. Suppose that for every y in R, there exists x such that df/dx (x,y) = 0. Suppose also that for each x in R there exists y such that df/dy (x,y) = 0. Can I prove that function f has a maximum under these assumptions.

Remark: the previous problem is correct for functions of the form f(x,y)=g(x)*h(y) and f(x,y)=g(x)+h(y).