Horizontal lines as local max/mins?

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lu6cifer
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Horizontal lines as local max/mins?

Postby lu6cifer » Sun Nov 08, 2009 7:53 pm UTC

Let's say you have a function with a horizontal line y = 5, which is also as the highest y-value. Let's say that y = 5 between x = [-4,4]. Could you say that a local maximum exists for the x-value of any point within [-4,4]? Or do local maxes and mins exist only for values that are a single point?
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Re: Horizontal lines as local max/mins?

Postby Qaanol » Sun Nov 08, 2009 7:55 pm UTC

A constant function has a local maximum at each point.
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lu6cifer
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Re: Horizontal lines as local max/mins?

Postby lu6cifer » Sun Nov 08, 2009 7:59 pm UTC

Well, I guess, more specifically, I'm talking about when f ' (x) bends down to touch the x-axis, doesn't cross it, and begins to increase again. At that point, this would create a horizontal line for the graph of f(x), right? Would that point be the x coordinate of a local maximum?
lu6cifer wrote:"Derive" in place of "differentiate" is even worse.

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RogerMurdock
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Re: Horizontal lines as local max/mins?

Postby RogerMurdock » Sun Nov 08, 2009 9:23 pm UTC

lu6cifer wrote:Well, I guess, more specifically, I'm talking about when f ' (x) bends down to touch the x-axis, doesn't cross it, and begins to increase again. At that point, this would create a horizontal line for the graph of f(x), right? Would that point be the x coordinate of a local maximum?


No, a local extrema is found when f'(x) changes from + to - or vice versa. Think about it, if f'(x) is never negative, that means the original function is never decreasing, and therefore there is no maximum. It's very possible to have functions like this, x^3 is one example. Look, it has a horizontal tangent line at x=0, but it is constantly increasing.

Image

GManNickG
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Re: Horizontal lines as local max/mins?

Postby GManNickG » Sun Nov 08, 2009 9:58 pm UTC

Off topic, but what did you use to graph/generate that image?

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Eebster the Great
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Re: Horizontal lines as local max/mins?

Postby Eebster the Great » Sun Nov 08, 2009 10:16 pm UTC

It sounds like you are trying to ask if f(x) has a local extremum at every x where f'(x) = 0. The answer is no, since there are plenty of counterexamples like the graph Roger showed you (f(x) = x^3). Instead, according to the first derivative test, if f is continuous on some interval I containing a, and f'(x) < 0 for x < a in I and f'(x) > 0 for x > a in I, then f has a relative minimum at a (it has a relative maximum if vice-versa). Note that this means either f'(a) = 0 or f'(a) doesn't exist. Points where f'(x) = 0 or f'(x) does not exist are called critical points.

If f is not differentiable on I (excluding a), things get more complicated. I'm actually not sure if there is any easy way to find relative extrema in most of these cases.

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RogerMurdock
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Re: Horizontal lines as local max/mins?

Postby RogerMurdock » Sun Nov 08, 2009 10:22 pm UTC

GManNickG wrote:Off topic, but what did you use to graph/generate that image?


I typed x^3 into wolframalpha.com and rehosted it on imageshack (wasn't sure i could link directly to the image that was hosted on wolfram).


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