Are intervals considered local min/max values of a function? If you have a function with an interval.
i.e. $y = 5x$ where $0 \leq  x \leq 5$
Obviously the function itself has no local min/max, but if the function is only for the interval between $0$ and $5$, are $x = 0$ and $x = 5$ considered local min/max values?
Sometimes I'm told it's $f'(x) = 0$, others it's whenever $f(x)$ is bigger or smaller than all the other values of $f(x)$ around it in an interval? Since $-1$ and $6$ are not defined since the function is only for the interval between $0$ and $5$, would that make them local min/maxima?

When testing for minimums and maximums, we check critical points, and included in these critical points are endpoints of intervals, in the case of a function defined on such an interval.
In your function, $y' = 5 \neq 0$ for all $x$. 
That leaves us with only the endpoints of the interval on which $y$ is defined as possible candidates for extrema. Since $y$ is always increasing, we can see that there is a minimum at $(0, 0)$ and a maximum  at $(5, 25).$ 
$y = f(x)$ is an extrema (minimum or maximum) only if $x$ is an element within the prescribed domain.
