# First order conditions for maximum likelihood estimator

Given a distribution $f(x_i,y_i;\alpha,\beta)$ and joint probability function $F(x_1,...,x_N,y_1,...,y_N;\alpha, \beta)$

The first order derivatives are $\frac{\partial F}{\partial \alpha}$ and $\frac{\partial F}{\partial \beta}$

Is the first order condition

$\frac{\partial F}{\partial \alpha}=0$ and $\frac{\partial F}{\partial \beta}=0$

OR

$\frac{\partial F}{\partial \alpha}=0$ or $\frac{\partial F}{\partial \beta}=0$

Do they need to satisfy at the same time or just one of them?