Is the following graph having two local minima

Question

https://www.desmos.com/calculator/abuvb1zdkb

I think yes, the main question i think is of the definition of neighbourhood

For a function with domain $(-\infty, -3)\cup (3, \infty)$

$ $

Is -3 in neighbourhood of 3 ?!

Answer 1

My "baby Rudin" (Principles of Mathematical Analysis, 2d ed), gives this definition on page 28 (where $d(p,q)$ means the distance between points $p$ and $q$ in a metric space).

A neighborhood of a point $p$ is a set $N_r(p)$ consisting of all points $q$ such that $d(p,q)<r$. The number $r$ is called the radius of $N_r(p)$.

So, for $r>6$, $-3$ is in the neighborhood $N_r(3)$; otherwise, not. So $-3$ is in some neighborhoods of 3 but not in all.

I note that $-3$ and $3$ are actually in the domain of the graph to which you linked. So, as I teach in my Calculus class, your graph has minima near $-3$ and $3$ which are both relative and absolute.