Local max and min at a if lim f(x) approaching a does not exist

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Local maxima and minima can exist at endpoints of a function defined over a finite interval, contrary to the high school perspective that requires the function to exist on both sides of the point. The discussion highlights a difference in definitions between high school and university teachings regarding local extrema. While it's common to focus on interior points for pedagogical clarity, endpoints can indeed serve as local extrema if the function is not constant near those points. This clarification emphasizes the importance of understanding the context in which local extrema are defined. Ultimately, local extrema can occur at endpoints, challenging traditional views on the subject.
merry
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Hi,
I understand that the local max of a function is the point at which the y value of the function is larger than the neighbouring y values of the function.
Say we're considering the local max at a of a function f(x). Does the function have to exist on both sides of a for (a,f(a)) to be a local max? (consider the same situation for a local min).
In short, if the fuction exists at [a, b] \epsilon\Re only,
can there be a local max or min at (a, f(a)) or (b, f(b)) ?
My high school professor said that to have local extrema, the function should exist on either side of the point. However, I believe my University prof said that this is not the case.
Could someone please clarify as to which one is the case?
Thanks a ton!
Merry
 
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It appears to be a quibble over definition. If you have a function defined over a finite interval, the end point values will almost always be local extrema (unless the function is constant near the end point). For pedogogical reasons, it is better to consider only interior points.
 
mathman said:
It appears to be a quibble over definition. If you have a function defined over a finite interval, the end point values will almost always be local extrema (unless the function is constant near the end point). For pedogogical reasons, it is better to consider only interior points.

Thanks! =D
I asked my Uni prof and he said that our high school textbooks are wrong and that local extrema can exist at end points xD
Thanks for the answer though!
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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