What would $\min(f(x))$ do?

Question

Would $\min(f(x))$ give one the minimum extremum of the function, or, would it give the $y$-value of the function with the smallest $x$-value possible? Or would it do neither? Is this notation even allowed?

Answer 1

Altough not so common, $\min(f(x))$ would mean the minimum value of $f(x)$, i.e smallest $y$-value as you called it, while $\underset{x}{\operatorname{argmin}} (f(x))$ is the x value that will give you that smallest value. You can think of it as a key-value relation. Though keep in mind, that when in doubt, be specific and write exactly what you mean.