Application of the Extreme Value Theorum

Question

I understand what the Theorem means and how to apply it effectively. For example, the theorem holds true at intervals of $f(x)= \sin(x)$ at $[-2,2]$. But one thing is confusing me is if the maxima AND minimum values exist on the END points of the interval. For example, if I said that at $$f(x): \sin(x) for [\frac{-\pi}{2},\frac{\pi}{2}]$$Would the theorem still apply? The way I have been taught that it shouldn't because it is at the boundaries, but what is the point of the $[\ ]$, shouldn't it say $( \ )$ or was I taught wrong?

Answer 1

The [ ] means the boundaries are included in looking for the maximum and minimum values. Don't get confused. The way the theorem is formulated is: $$ f(c) \geq f(x) \geq f(d), \forall x \in [a,b] $$ Note that a can be equal to c or d, and b can be equal to c or d.