Is it a must that all critical points are interior point for a function ?
My question is can a boundary point be critical point of a function??
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Endpoints are not generally considered to be differentiable (because no two sided limits). – Andrew Li Feb 27 '18 at 13:23
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That's a great question that a student of mine once raised, and I realized that I had never seen any calculus book, or even analysis book, that addressed the question. On the one hand, if your function is defined on a closed interval, the two-sided derivative doesn't technically exist at the boundary points. On the other hand, it doesn't seem quite right to say that the function $f(x)=x^2$ isn't differentiable on the interval $[0,1]$, since the function obviously extends to any interval we want. What's the way out? As I understand it, boundary points are never critical points, and that is by definition. When you're doing the optimization strategy of finding all the critical points, you just always check the boundary points as an additional matter of course.
Adrian Keister
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No problem! I think most calculus books aren't expecting their students to think this deeply about the matter. – Adrian Keister Feb 27 '18 at 14:25