Formulation of a sufficient condition for extrema

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# Formulation of a sufficient condition for extrema

Submitted by Philidor on Sun, 08/07/2011 - 12:45

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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.

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## Re: Formulation of a sufficient condition for extrema

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