Only Critical Point in Town
For a function of one variable there is a test called the "only critical point in town." It says that if a differentiable function has only one critical point and it is a local maximum (or minimum), then it is a global maximum (or minimum). You can prove this using Rolle's Theorem.
However, the "only critical point in town" test does not hold for functions of two variables. Here we see two examples. Both functions have only one critical point each. The blue function has a local minimum at (0,0) that is not a global minimum, and the purple function has a local maximum at (1,0) that is not a global maximum.