Point of local extremum implies critical point - Revision history

Vipul: /* General comments */

2024-05-13T06:12:28Z

General comments

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	* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.		* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
	* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of ~~discontunity~~ (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>. With that being said, in many cases, we can rule out limiting behaviors of that sort, and the set of points of local extremum is small enough that it's feasible to find the absolute extremum by looking among them.		* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontinuity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>. With that being said, in many cases, we can rule out limiting behaviors of that sort, and the set of points of local extremum is small enough that it's feasible to find the absolute extremum by looking among them.

	===Qualitative and existential significance===		===Qualitative and existential significance===

Vipul: /* General comments */

2024-04-18T06:49:36Z

General comments

← Older revision		Revision as of 06:49, 18 April 2024
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	* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.		* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
	* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.		* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>. With that being said, in many cases, we can rule out limiting behaviors of that sort, and the set of points of local extremum is small enough that it's feasible to find the absolute extremum by looking among them.

	===Qualitative and existential significance===		===Qualitative and existential significance===

Vipul at 06:48, 18 April 2024

2024-04-18T06:48:35Z

← Older revision		Revision as of 06:48, 18 April 2024
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	Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.		Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

			==Related facts==

			* [[Point of local extremum implies critical point for a function of multiple variables]]

	==Significance==		==Significance==

Vipul: /* Computational feasibility significance */

2024-04-18T06:47:22Z

Computational feasibility significance

← Older revision		Revision as of 06:47, 18 April 2024
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Root-finding method !! ~~Optimization~~ method		! Root-finding method !! Corresponding optimization method
	\|-		\|-
	\| [[Newton's method for root-finding for a function of one variable]] \|\| [[Newton's method for optimization of a function of one variable]]		\| [[Newton's method for root-finding for a function of one variable]] \|\| [[Newton's method for optimization of a function of one variable]]

Vipul: /* Significance */

2024-04-18T06:46:46Z

Significance

← Older revision		Revision as of 06:46, 18 April 2024
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	In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.		In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

			From the perspective of numerical approximation, any method for numerically approximating the ''roots'' of a function in a family <math>\mathcal{F}</math> can become a method for finding the critical points (and hence, the points of local extremum) of a function whose derivative is in <math>\mathcal{F}</math>. Some of these translations are below:

			{\| class="sortable" border="1"
			! Root-finding method !! Optimization method
			\|-
			\| [[Newton's method for root-finding for a function of one variable]] \|\| [[Newton's method for optimization of a function of one variable]]
			\|}

	==Facts used==		==Facts used==

Vipul: /* Significance */

2024-04-18T06:42:39Z

Significance

← Older revision		Revision as of 06:42, 18 April 2024
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	\| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| <math>d_1 + d_2 - 1</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. \|\| <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.		\| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| <math>d_1 + d_2 - 1</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. \|\| <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
	\|}		\|}

			===Computational feasibility significance===

			As a general rule, for any family of functions <math>\mathcal{F}</math> for which we know how to find the roots of any function in the family (i.e., the points where the function is zero), we have a strategy to find critical points (and therefore points of local extremum) for any function whose ''derivative'' is in <math>\mathcal{F}</math>.

			In particular, since we know how to find the roots of polynomials of degree 1 (linear polynomial) or 2 (quadratic polynomial), we can find the critical points of any degree 2 (quadratic) or degree 3 (cubic) function, as well as of any rational function with a linear numerator and quadratic denominator.

	==Facts used==		==Facts used==

Vipul: /* Qualitative and existential significance */

2024-04-18T06:38:32Z

Qualitative and existential significance

← Older revision		Revision as of 06:38, 18 April 2024
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Type of function for which we are interested in understanding local extrema !! Inference		! Type of function for which we are interested in understanding local extrema !! Maximum number of points of local extremum !! Inference !! Example
	\|-		\|-
	\| polynomial function of degree <math>d</math> \|\| The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.		\| polynomial function of degree <math>d</math> \|\| <math>d - 1</math> \|\| The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function. \|\| <math>x^3 - 3x</math> of degree <math>d = 3</math> has <math>3 - 1 = 2</math> critical points: the two solutions to <math>3x^2 - 3 = 0</math>, namely <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
	\|-		\|-
	\| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum.		\| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| <math>d_1 + d_2 - 1</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum. \|\| <math>x/(x^2 + 1)</math> with <math>d_1 = 1, d_2 = 2</math> has derivative <math>(1 - x^2)/(x^2 + 1)^2</math> and has <math>1 + 2 - 1 = 2</math> critical points, namely the solutions to <math>1 - x^2 = 0</math>, so <math>x = -1</math> and <math>x = 1</math>. Both of these turn out to be points of local extremum.
	\|}		\|}

Vipul: /* Significance */

2024-04-18T06:34:37Z

Significance

← Older revision		Revision as of 06:34, 18 April 2024
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	\| polynomial function of degree <math>d</math> \|\| The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.		\| polynomial function of degree <math>d</math> \|\| The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.
	\|-		\|-
	\| rational function with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2</math> and denominator a square of a polynomial of degree <math>d_2</math>, so there are at most <math>d_1 + d_2</math> critical points ~~of the "derivative is zero" type~~ and at most <math>d_2</math> ~~critical~~ points of ~~the "derivative is undefined" type. This puts an upper bound of <math>d_1 + 2d_2</math> on the number of critical points~~.		\| rational function in simplified form with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2 - 1</math> and denominator a square of the original denominator. The only way for the derivative to be undefined is for the denominator to be zero, in which case the denominator of the original rational function would also be zero, so that the original rational function would also be undefined. Therefore, the derivative exists everywhere that the function is defined, and the only critical points are cases where the derivative is zero. Since the numerator is a polynomial of degree at most <math>d_1 + d_2 - 1</math>, there are therefore at most <math>d_1 + d_2 - 1</math> critical points, and therefore at most <math>d_1 + d_2 - 1</math> points of local extremum.
	\|}		\|}

Vipul at 06:29, 18 April 2024

2024-04-18T06:29:12Z

← Older revision		Revision as of 06:29, 18 April 2024
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	Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.		Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

			==Significance==

			===General comments===

			The main significance of this result is that it mostly reduces an ''optimization'' problem (finding a maximum or minimum) to an ''equation-solving'' problem (finding when an expression is zero).

			The reduction isn't perfect, in that it has the following caveats:

			* The result is only a one-sided implication. Any point of local extremum is a critical point, but it's possible for a critical point to not be a point of local extremum. Even so, in many cases, the set of critical points is already small enough and manageable enough.
			* This result is about local extrema, not absolute / global extrema. There may be many points of local extremum that aren't points of absolute extremum. Moreover, there could be many cases where an absolute extremum doesn't exist; for instance, for a function that is going to infinity as the input goes to infinity or to a point of discontunity (such as <math>x \mapsto 1/x</math> as <math>x \to 0^+</math> and <math>x \to 0^-</math>.

			===Qualitative and existential significance===

			{\| class="sortable" border="1"
			! Type of function for which we are interested in understanding local extrema !! Inference
			\|-
			\| polynomial function of degree <math>d</math> \|\| The derivative is a polynomial of degree <math>d - 1</math>, so it is defined everywhere and has at most <math>d - 1</math> real roots. Therefore, there are at most <math>d - 1</math> critical points (all of the "derivative equals zero" type, none of the "derivative is undefined" type) and therefore at most <math>d - 1</math> points of local extremum for the function.
			\|-
			\| rational function with numerator degree <math>d_1</math> and denominator degree <math>d_2</math> \|\| The derivative is a rational function with numerator degree at most <math>d_1 + d_2</math> and denominator a square of a polynomial of degree <math>d_2</math>, so there are at most <math>d_1 + d_2</math> critical points of the "derivative is zero" type and at most <math>d_2</math> critical points of the "derivative is undefined" type. This puts an upper bound of <math>d_1 + 2d_2</math> on the number of critical points.
			\|}

	==Facts used==		==Facts used==

Vipul: /* Statement */

2024-04-18T06:12:42Z

Statement

← Older revision		Revision as of 06:12, 18 April 2024
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	Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.		Then, <math>c</math> is a [[critical point]] for <math>f</math>, i.e., either the [[derivative]] <math>f'(c)</math> equals zero or the derivative <math>f'(c)</math> does not exist.

	Note that the <math>\! f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.		Note that the <math>f'(c)</math> not existing case could occur in either of these ways: one or both the one-sided derivatives of <math>f</math> at <math>c</math> not existing, ''or'' both one-sided derivatives of <math>f</math> at <math>c</math> existing but being unequal.

	==Facts used==		==Facts used==