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mcastillo356
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- TL;DR Summary
- I've got a proof of the inexistence of a local maximum value nor a local minimum value at the origin of coordinates for a certain function, and need advice to understand it
Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value
A function with no max or min at an endpoint. Let
Show that is continuous on and differentiable on but it has neither a local maximum nor a local minimum value at the endpoint
I've already proven it is continuous on and differentiable on , and here is the proof that it has not loc min or loc max:
Background
A function has not a maximum local value at in its domain if for all can always be found , such that and
A function has not a minimum local value at in its domain if for all can always be found , such that and
Proof I don't understand:
For all we have and , therefore .
Let .Exists one such that . Then for all we have:
. Then and
No matter how small is we have . And is proven that cannot be maximum nor minimum.
A function with no max or min at an endpoint. Let
Show that
I've already proven it is continuous on
Background
A function
A function
Proof I don't understand:
For all
Let
No matter how small is