No. of critical points

Question

Domain: $\mathopen]1,4\mathclose[$, $f(x)= 3x^2 - 6x$ How many critical points exist? (Zero, 1, 3, 4)

By diff.($x$): $f'(x)= 6x-6$ Then $x=1$ is local point of minimum, but according to the interval should I consider it or not as a critical point?

Answer 1

A function maps a domain (a set) to another set. A function is not only defined by how it maps points but also by WHICH points (the domain) are mapped and indicating what the domain is an essential aspect of defining a function.

That means the function $f: ]1,4[ \to \mathbb R$ via $f(x) = 3x^2 - 6x$ is a DIFFERENT function than $\overline{f}:\mathbb R\to \mathbb R$ via $\overline{f}(x) = 3x^2 - 6x$.

So $x = 1$ can not be a critical point of $f$ because $x = 1$ is not a point in $f$'s domain at all.