Relative extrema standard In a national level exam for university admissions, we were asked to find the number and location(s) of the local minima of $\sin(x)$ , in $[\pi/4,7\pi/4]$ .
$3\pi/2$ is surely one point of local minima, as points about $3\pi/2$ have larger values for $\sin(x)$.
However, there seems to be a confusion for $\pi/4$ . Some definitions of local minima say that they cannot occur at the endpoints of the domain considered, while other definitions say that endpoints also qualify for  having the local extrema. In this case, values of $\sin(x)$ to the immediate right of $x = \pi/4$ are bigger.
In this particular exam, the accepted answer was $\pi/4$ and $3\pi/2$ (2 points), which means endpoints were considered.
Is there an agreed upon definition for this? Different authors have different opinions about this.
Can someone clarify this?
 A: Given a space $X$  and a point $\xi\in X$ the function $f:\>X\to{\mathbb R}$ is locally minimal at $\xi$ if there is a neighborhood $U$ of $\xi$ such that
$f(x)\geq f(\xi)$ for all $x\in U$.
In the example at hand $X:=\bigl[{\pi\over4},{7\pi\over4}\bigr]\subset{\mathbb R}$ and 
$$f(x):=\sin x\quad(x\in X)\ .$$
By inspection of the graph of $f$ we see that there are two local minima, namely the points $\xi_1={\pi\over4}$ and $\xi_2={3\pi\over2}$.
For a less known $f$ one would have to find all zeros of $f'$ in the interior of the given interval $X$ and check each of them using the second derivative test, whether it is a local minimum; in addition one would have to analyze $f$ in the vicinity of the boundary points of $X$.
A: Suppose that the problem had be : minimize Sin(x) over the range [Pi/4 , 7 Pi/4]. Plot your function and you see that on the left hand side, Pi/4 leads to a constrained minimum and 3 Pi/2 to an unconstrained minimum.   
Does this clarify anything for you ? If not, please post.
A: It seems that your question asks you to find the set of local minima of the function $\sin(x)$ over $\Bbb{R}$, and then restrict it to $[\pi/4,7\pi/4]$. If you are instead considering the set of local minima of the function $\sin(x)|_{[\pi/4,7\pi/4]}$, the restriction of $\sin(x)$ to $[\pi/4,7\pi/4]$, you may get a different result if you decide to exclude the values at the endpoints in your definition of local minima. 
