Ellie C.

asked • 02/07/18

Which of the following must be true at a local maximum?

For a differentiable function, which of the following must be true at a local
maximum?
I f'(x)=0
II f'(x) changes sign
III f"(x) = 0
IV f"(x) changes sign

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Bobosharif S. answered • 02/07/18

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Local maximum is a point x0, where f'(x0)=0. This means that up to the point x0 function f(x) increases, f'(x)>0, for x<x0 and decreases f'(x)<0 for x>x0.  f'(x) changes passing the point x0. Now you figure out which answer should be correct.
 
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Michael W.

Sorry, but I think there's a technicality here that the original response doesn't reflect.
 
A local extreme occurs at a place where the derivative of a function is either 0...or undefined, and where the derivative changes sign.  The graph of the absolute function (which looks like a V) is the classic example of a function that clearly has a local minimum (when x = 0, at the tip of the V), but the derivative isn't zero at that point.  It's undefined, because of the sharp turn.
 
So, based on the more specific definition, I think you'd get a different answer for which statements absolutely positively must be true.  Roman Numeral I is the tricky one.
 
So,
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02/08/18

Bobosharif S.

Yes, you are right. I din't give you an answer. I'm just trying to show how you can answer the question yourself. I gave you, say, some hints.  And it is good to hear that you mentioned about derivative being indefinite.
Variants of your answers are "f'(x)=0" (twice) and  "f"(x)  changes sign" (again twice), which basically both are true. If f'(x0) =0 and f'(x) has different signs "before" and "after" x0, that is, if it changes sign, then x0 is an extreme point. To get a final answer, you have to clarify, will f'(x) change sigh if f'(x0) indefinite.
 
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02/08/18

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