Extrema - Revision history

DavidB4-bot: /* Problem Solving Methodology */clean up & uniformity

2016-07-13T12:18:10Z

Problem Solving Methodology: clean up & uniformity

Aschlafly: improved

2009-12-25T05:22:57Z

improved

Aschlafly: moved Maxima to Extrema: per suggestion

2009-12-25T05:21:50Z

moved Maxima to Extrema: per suggestion

Aschlafly: Problem Solving Methodology

2009-12-25T05:21:13Z

Problem Solving Methodology

JimR at 04:17, December 25, 2009

2009-12-25T04:17:56Z

JacobB at 04:11, December 25, 2009

2009-12-25T04:11:09Z

JimR: comment on compact spaces and maxima, with a hopefully useful example. please remove if this section isn't helpful.

2009-12-25T03:59:53Z

comment on compact spaces and maxima, with a hopefully useful example. please remove if this section isn't helpful.

Aschlafly: improved

2009-12-25T03:52:26Z

improved

JimR: expand definition, add examples. if someone with upload rights could add a graph, it would be a nice addition

2009-12-25T03:51:04Z

expand definition, add examples. if someone with upload rights could add a graph, it would be a nice addition

Aschlafly: for multi-variable functions

2009-12-25T03:44:58Z

for multi-variable functions

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 :fourth, plug in the various critical points for (x,y) back into ''f(x,y)'' and simply identify the absolute and relative maxima and minima.
-[[Category:calculus]]
+[[Category:Calculus]]
-[[Category:mathematics]]
+[[Category:Mathematics]]

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-{{move|Extrema}}
+'''Maxima''' and '''minima''' are a type of '''extrema''' or critical points (numbers) for functions, which can be found with calculus.
 The ''absolute'' maximum of a function is the point in the domain at which the function achieves its largest value.  For example, the absolute maximum of <math>f(x) = 1- x^2</math> occurs at <math>x=0</math>, where <math>f(0) = 1</math>.
@@ Line 11: / Line 10: @@
 To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.
-The [[minima]] of a function can be calculated in an analogous manner.
+The '''minima''' of a function can be calculated in an analogous manner.
 The computation of maxima of functions remains an important issue in more advanced mathematics.  For example, it is often useful to consider real-valued continuous functions on more exotic topological spaces, and ask what their maxima are.  Note that some functions do not have maxima: for example, <math>f(x) = x^2</math> takes on arbitrarily large values.  A fundamental theorem states that any function defined on a [[compact space]] obtains a maximum value; this is just a consequence of the fact that the image of a compact space under a continuous function is compact, and thus bounded.

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 A simple example of this distinction is the following: one can define a function on the surface of the earth, whose value at a point is the current temperature at that point.  One could similarly define such a function on the entire universe, sending a location to its temperature.  Since the earth is compact, it follows that there is some point on earth which is the hottest at any given time.  However, if the universe is unbounded, it is not compact, and we can not reach the same conclusion!  There may be points in the universe which have arbitrarily large temperatures, so that the function has no maximum.
 [[Category:calculus]]
 [[Category:mathematics]]

← Older revision		Revision as of 04:17, December 25, 2009
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	The ''relative'' maximum of a function is a point which at which the function takes on its largest value on some neighborhood of the point, i.e., the function takes on smaller values at all sufficiently close points. For example, the function <math>f(x) = e^{-x^2} \cos x</math> has many relative maxima, and a unique absolute maximum at <math>x=0</math>. A relative maximum of a function can occur when its first derivative is zero and its second derivative is less than zero or, alternatively, its first derivative is positive approaching the point from the left and negative to its right.		The ''relative'' maximum of a function is a point which at which the function takes on its largest value on some neighborhood of the point, i.e., the function takes on smaller values at all sufficiently close points. For example, the function <math>f(x) = e^{-x^2} \cos x</math> has many relative maxima, and a unique absolute maximum at <math>x=0</math>. A relative maximum of a function can occur when its first derivative is zero and its second derivative is less than zero or, alternatively, its first derivative is positive approaching the point from the left and negative to its right.
		+
		+	[[Image:Relmaxplotsmall.jpg\|center\|The graph of a function with numerous relative maxima]]

	To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.		To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.

← Older revision		Revision as of 04:11, December 25, 2009
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		+	{{move\|Extrema}}
	'''Maxima''' are a type of extrema or critical number for functions, which can be found with calculus.		'''Maxima''' are a type of extrema or critical number for functions, which can be found with calculus.

← Older revision	Revision as of 05:21, December 25, 2009
(No difference)

← Older revision		Revision as of 03:52, December 25, 2009
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	To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.		To find the '''maxima''' of multi-variable functions, take the partial derivatives of the function with respect to each variable and set each of those resulting equations to zero.
		+
		+	The [[minima]] of a function can be calculated in an analogous manner.
		+	[[Category:calculus]]
	[[Category:mathematics]]		[[Category:mathematics]]
−	~~[[Category:calculus]]~~