Notation for greatest value a variable can be

Question

I'd like to know if there is a notation for a term or pronumeral is the highest value it can be (usually in terms of an equation or equality)

For Example:

Pretend "??" is such notation. $$y=-x^2+1$$$$y??$$ What is the value of y? The answer is obviously $y=1$ however is there a correct notation for this.

Answer 1

This is simply $\max(y)$, the maximum of $y$.

To say $m$ is the largest value, we say $$m=\max_{x\in\mathbb R}\{1-x^2\},$$ i.e., "$m$ is the maximum over real $x$ of the set of numbers of the form $1-x^2$".

Answer 2

The maximum of a function is denoted as

$$\max_xf(x),$$ where the lower $x$ indicates on which variable(s) you optimize, and possibly in what range.

The notation

$$\arg\max_xf(x)$$ denotes the value of the variable for which that maximum is reached.