Best way for defining global maximum and minimum for any Interval in a function

Question

Let $a > 0$ be some $\mathbb{R}$ . Find the global maximum and the global minimum of the function: $f(x) = x^3 - x^2 $ on the interval $[-a,a]$.

Is it valid to just define a set of all possible y values: $ Y := $ {$ x \in [-a,a]: f(x$} and then create a set of a single global x value in the following: global maximum := {$ !\exists x \in [-a,a], \forall y \in Y: f(x) \geq y$} and the same for the global minimum := {$ !\exists x \in [-a,a], \forall y \in Y: f(x) \leq y$}?

Would this be a mathematically correct answer? If yes, is there a more elegant way? Thank you very much.

Answer 1

It is mathematically correct. It is useless. You are given an explicit $f$; you should be able to do vastly better than just recite the definition of global minimum and global maximum. For instance, with this $f$, the global minimum on $[-a,a]$ for $a \geq 0$ is $-a^3 - a^2$. The global maximum is a piecewise function with two pieces.

Answer 2

From Wikipedia:

A real-valued function f defined on a domain X has a global (or absolute) maximum point at $x∗$ if $f(x∗) ≥ f(x)$ for all $x \in X$. Similarly, the function has a global (or absolute) minimum point at $x∗$ if $f(x∗) ≤ f(x)$ for all $x \in X$.

What you call $Y$ is the codomain of $f$. Formally speaking, $f : X \to Y$ has global minimum $x_0 \in X$ on the interval $X$ if $(\forall x \in X) (f(x_0) \geq f(x))$. The definition of global minimum is analogous.