I study for the first time the problem of finding saddle points and I have a question. So, the problem says that
Suppose that $A \subset \mathbb{R}^n$, $B \subset \mathbb{R}^m$ are subsets of Euclidean space and $f(x,y)$ a numerical function given on $\mathbb{R}^n \times \mathbb{R}^m$. We want to find the point $(x^*, y^*) \in A \times B$ such that $$f(x^*, y) \leq f(x^*, y^*) \leq f(x, y^*), \forall x \in A, y \in B$$
What I don't understand is why this problem is equivalent with the following
The point $(x^*, y^*) \in A \times B$ is a saddle point if and only if the inequalities $$<f_x (x^*, y^*), x - x^*> \geq 0, \forall x \in A \\ <f_y (x^*, y^*), y - y^*> \leq 0, \forall y \in B$$ hold.
I hope you can clear me up. Thanks!