A potentially naïve question about finding the minimum of a function:
If I have a scalar function of a scalar variable $y=f(x)$, which is continuously differentiable, I can find minima but finding the values of $x$ for which $f'(x)=0$.
On the other hand, if I have a scalar function of a vector variable, $y=f(\mathbf{x})$, can I do the same? Can I always find the value of $\mathbf{x}$ where $f'(\mathbf{x})=0$?
I am aware I can take partial derivatives for each value of $\mathbf{x}$, i.e.
$\nabla y = \left[\frac{\partial y}{\partial x_1}, \frac{\partial y}{\partial x_2}, \dots \frac{\partial y}{\partial x_N} \right]$
But how do I go from this vector of derivatives to a final value for $\mathcal{x}$? Do I just set each derivative to zero?