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Formulation of a sufficient condition for extrema

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Formulation of a sufficient condition for extrema

In many books the necessary condition of f'(x)=0 is put into the sufficient condition for a function having an extremum at x, for example in the following way:

f has an extremum at x, if
1) f'(x)=0 and
2) f'(x) changes its sign when passing x

Now, isnt this sufficient condition overloaded? Would it not also be sufficient to ask for:
1*) f is differentiable at x
2) f'(x) changes its sign when passing x

It seems to me that f'(x)=0 can be concluded from this sufficient condition and does not need to be built into it.

Thx for an affirmation or a counterexample/Philidor.


It is true. But this is due to the Darboux Theorem, which states that if f(x) is a differentiable function, then f'(x) is a Darboux function, i.e. it has the intermediate value property (not necessarly continous). So if we do the extremum test for x=a and if for x<a we have f'(x)<0 and for x>a we have f'(x)>0 (not for all x<a or x>a but in some small neighbourhood), then f'(a)=0 because of the intermediate value property.

joking

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