Suppose that for some function $f$ we want to know the stationary points, i.e. $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$.
We can define a new function $\widetilde{f}(\mathbf{x}) = \left( \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \right)^2$, which is zero when the partial derivatives w.r.t. each $x$ is zero. For this function we have that $\frac{\partial \widetilde{f}(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{2}^T \frac{\partial^2 f(\mathbf{x})}{\partial \mathbf{x}^2} \text{diag} \left( \frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} \right)$
The partial derivative w.r.t. a single variable is given by $\frac{\partial \widetilde{f}(\mathbf{x}}{\partial x_i} = \frac{\partial f(\mathbf{x})}{\partial x_i} \sum\limits_j \frac{\partial^2 f(\mathbf{x})}{\partial x_i \partial x_j}$
The stationary points of $f$ are then also stationary points of $\widetilde{f}$, since when $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$ it holds that $\frac{\partial \widetilde{f}(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$
Sometimes this may help in finding an analytic solution for $\mathbf{x}$ such that $\frac{\partial f(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{0}$
Is there a name for this kind of process? Is this a generally accepted technique?