I am trying prove that a function has a unique optimal solution. It is a function with two variables, and it is not necessarily well-behaved, in particular is not concave. Can I argue the following?

The vector of first order derivatives is decreasing in the vector of variables, hence the function is unimodal and would have a unique optimal solution.

In other words, for univariate functions, we can show the unimodality by argueing that the first derivative is decreasing. Can I generalize it to the multivariate case as described above?

Thanks for any help provided.