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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?

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    Yes, that is a necessary condition for extremum for $C^1$ functions on $\mathbb R^n$: $\frac{\partial y}{\partial x_i}(\hat{\mathbf x}) = 0$ for all $i=1,\dots,n$ whenever $\hat{\mathbf x}$ is an extremum argument of the function $y$.2012-08-01

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