Quadratic function - Revision history

Vipul: /* Generalizations */

2014-05-26T06:48:09Z

Generalizations

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	==Generalizations==		==Generalizations==

	* [[Convex function]] ~~generalized~~ quadratic function with positive leading coefficient		* [[Convex function]] generalizes quadratic function with positive leading coefficient
	* [[Quadratic function of multiple variables]]		* [[Quadratic function of multiple variables]]
	* [[Polynomial function]]		* [[Polynomial function]]

Vipul: /* Definition */

2014-05-26T04:36:11Z

Definition

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	Unless otherwise specified, we consider quadratic functions where the inputs, outputs, and coefficients are all real numbers.		Unless otherwise specified, we consider quadratic functions where the inputs, outputs, and coefficients are all real numbers.

			==Generalizations==

			* [[Convex function]] generalized quadratic function with positive leading coefficient
			* [[Quadratic function of multiple variables]]
			* [[Polynomial function]]

	==Related notions==		==Related notions==

Vipul: /* Local extreme values */

2014-05-26T03:38:28Z

Local extreme values

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	The second derivative is the constant function with value <math>2a</math>, which is never zero. Therefore, there are no points of inflection.		The second derivative is the constant function with value <math>2a</math>, which is never zero. Therefore, there are no points of inflection.

	===Local extreme values===		===Local and absolute extreme values===

	====Case a > 0====		====Case a > 0====

Vipul: /* Key invariants */

2014-05-26T03:37:47Z

Key invariants

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	\| <math>-b/a</math> \|\| sum of roots \|\| If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the sum of roots <math>\alpha + \beta</math> is <math>-b/a</math>.		\| <math>-b/a</math> \|\| sum of roots \|\| If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the sum of roots <math>\alpha + \beta</math> is <math>-b/a</math>.
	\|-		\|-
	\| <math>c/a</math> \|\| product of ~~rots~~ \|\| If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the product of roots <math>\alpha\beta</math> is <math>c/a</math>.		\| <math>c/a</math> \|\| product of roots \|\| If the roots are <math>\alpha,\beta</math> (counted with multiplicity, and they need not be real roots), then the polynomial is <math>a(x - \alpha)(x - \beta)</math> and the product of roots <math>\alpha\beta</math> is <math>c/a</math>.
	\|-		\|-
	\| <math>b^2/(4a^2) - c/a</math> \|\| normalized discriminant \|\| Similar observations as for the discriminant.		\| <math>b^2/(4a^2) - c/a</math> \|\| normalized discriminant \|\| Similar observations as for the discriminant.

Vipul: /* Local extreme values */

2014-05-26T03:34:15Z

Local extreme values

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	===Local extreme values===		===Local extreme values===

	====Case ~~<math>~~a > 0~~</math>~~====		====Case a > 0====

	In this case, the unique critical point <math>x = -b/(2a)</matH> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in any of the following equivalent ways:		In this case, the unique critical point <math>x = -b/(2a)</matH> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in any of the following equivalent ways:
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	\|}		\|}

	====Case ~~<math>~~a < 0~~</math>~~====		====Case a < 0====

	In this case, the unique critical point <math>x = -b/(2a)</math> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in three ways, analogous to the case <math>a > 0</math>.		In this case, the unique critical point <math>x = -b/(2a)</math> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in three ways, analogous to the case <math>a > 0</math>.

Vipul: /* =Case a > 0 */

2014-05-26T03:33:49Z

=Case a > 0

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	===Local extreme values===		===Local extreme values===

	====Case <math>a > 0</math>===		====Case <math>a > 0</math>====

	In this case, the unique critical point <math>x = -b/(2a)</matH> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in any of the following equivalent ways:		In this case, the unique critical point <math>x = -b/(2a)</matH> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in any of the following equivalent ways:
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	\| [[second derivative test]] \|\| The second derivative is the positive number <math>2a</math> everywhere, so the critical point is a point of local minimum, and also the unique point of absolute minimum. \|\| The second derivative being positive ''at the critical point'' suffices to deduce that it is a point of local minimum. The second derivative being positive everywhere tells us that the function is a [[convex function]] (i.e., the graph is concave up), and therefore, any local minimum is unique and equals the absolute minimum.		\| [[second derivative test]] \|\| The second derivative is the positive number <math>2a</math> everywhere, so the critical point is a point of local minimum, and also the unique point of absolute minimum. \|\| The second derivative being positive ''at the critical point'' suffices to deduce that it is a point of local minimum. The second derivative being positive everywhere tells us that the function is a [[convex function]] (i.e., the graph is concave up), and therefore, any local minimum is unique and equals the absolute minimum.
	\|}		\|}

			====Case <math>a < 0</math>====

			In this case, the unique critical point <math>x = -b/(2a)</math> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in three ways, analogous to the case <math>a > 0</math>.

	==Integration==		==Integration==

Vipul: /* Integration */

2014-05-26T03:32:47Z

Integration

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	Since the second derivative is a constant (as we find using either of the computational methods above), all higher derivatives are zero.		Since the second derivative is a constant (as we find using either of the computational methods above), all higher derivatives are zero.

			==Points and intervals of interest==

			===Critical points===

			We can determine the critical point by setting the derivative to equal zero:

			<math>2ax + b = 0</math>

			This solves to <math>x = \frac{-b}{2a}</math>.

			Alternatively, we can look at the form:

			<math>f(x) = a\left(x - \frac{-b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)</math>

			The critical point is therefore a shift by <math>-b/(2a)</math> of the critical point for the [[square function]], which is known to be at <math>0</math>.

			===Points of inflection===

			The second derivative is the constant function with value <math>2a</math>, which is never zero. Therefore, there are no points of inflection.

			===Local extreme values===

			====Case <math>a > 0</math>===

			In this case, the unique critical point <math>x = -b/(2a)</matH> defines a [[point of local minimum]], and it is also the unique point of absolute minimum. This can be seen in any of the following equivalent ways:

			{\| class="sortable" border="1"
			! Method !! Description !! Comment on deducing local versus absolute minimum
			\|-
			\| transformed description of function \|\| We have <math>f(x) = a\left(x - \frac{-b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)</math>. The constant term <math>\left(c - \frac{b^2}{4a}\right)</math> does not affect the point of local minimum. The expression <math>a\left(x - \frac{-b}{2a}\right)^2</math> is always nonnegative (because <math>a > 0</math>). It is zero iff <math>x - \frac{-b}{2a} = 0</math>, or <math>x = \frac{-b}{2a}</math>. This, therefore, is the unique point of local and absolute minimum. \|\| Note that this method directly gives us that it is the unique point of absolute minimum.
			\|-
			\| [[first derivative test]] \|\| The derivative function <math>x \mapsto 2ax + b = 2a\left(x - \frac{-b}{2a}\right)</math> is negative for <math>x < \frac{-b}{2a}</math> and positive for <math>x > \frac{-b}{2a}</math>. Therefore, <math>f</math> is decreasing on the left and increasing on the right of <math>-b/(2a)</math>, and thus has its unique local and absolute minimum at this point. \|\| The first derivative test, as conventionally stated, only talks about the sign of the derivative on the ''immediate'' left and right, and in this form, suffices only to deduce ''local'' extreme behavior. However, in this case, we have global information on the sign of derivative on the left and right of the point, therefore we can deduce that we have an absolute (global) minimum.
			\|-
			\| [[second derivative test]] \|\| The second derivative is the positive number <math>2a</math> everywhere, so the critical point is a point of local minimum, and also the unique point of absolute minimum. \|\| The second derivative being positive ''at the critical point'' suffices to deduce that it is a point of local minimum. The second derivative being positive everywhere tells us that the function is a [[convex function]] (i.e., the graph is concave up), and therefore, any local minimum is unique and equals the absolute minimum.
			\|}
	==Integration==		==Integration==

Vipul: /* Higher derivatives */

2014-05-26T03:19:28Z

Higher derivatives

← Older revision		Revision as of 03:19, 26 May 2014
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	Since the second derivative is a constant (as we find using either of the computational methods above), all higher derivatives are zero.		Since the second derivative is a constant (as we find using either of the computational methods above), all higher derivatives are zero.

			==Integration==

			===First antiderivative===

			We can use that [[indefinite integration is linear]] to convert this to a problem of integrating power functions:

			<math>\int (ax^2 + bx + c) \, dx = a \int x^2 \, dx + b \int x^1 \, dx + c \int x^0 \, dx</math>

			We can now use the [[integration rule for power functions]] (<math>\int x^r \, dx = \frac{x^{r+1}}{r + 1} + C</math>) and obtain:

			<math>\int (ax^2 + bx + c) \, dx = a \frac{x^3}{3} + b \frac{x^2}{2} + cx + C, C \in \R</math>

			===Higher antiderivatives===

			The process of integration can be repeated. The following is the formula for the <math>k</math>-fold integral of the function:

			<math>a \frac{x^{k+2}2!}{(k + 2)!} + b \frac{x^{k+1}}{(k + 1)!} + c \frac{x^k}{k!} + \mbox{arbitrary polynomial of degree at most } k - 1</math>

			The arbitrary polynomial of degree at most <math>k - 1</math> is parameterized by <math>k</math> coefficients. These are the constants of integration that accumulate, one in each step of integration.

Vipul at 03:12, 26 May 2014

2014-05-26T03:12:48Z

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	<math>\frac{d}{dx}(ax^2 + bx + c) = a \frac{d}{dx}(x^2) + b \frac{d}{dx} x + c \frac{d}{dx} 1</math>		<math>\frac{d}{dx}(ax^2 + bx + c) = a \frac{d}{dx}(x^2) + b \frac{d}{dx} x + c \frac{d}{dx} 1</math>

	Now, using the ~~rule for~~ [[differentiation of power functions]], namely <math>d(x^n)/dx = nx^{n-1}</math>, we obtain:		Now, using the [[differentiation rule for power functions]], namely <math>d(x^n)/dx = nx^{n-1}</math>, we obtain:

	<math>\frac{d}{dx}(ax^2 + bx + c) = a (2x) + b(1) = 2ax + b</math>		<math>\frac{d}{dx}(ax^2 + bx + c) = a (2x) + b(1) = 2ax + b</math>
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	<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = a \frac{d^2}{dx^2}(x^2) + b \frac{d^2}{dx^2} x + c \frac{d^2}{dx^2} 1</math>		<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = a \frac{d^2}{dx^2}(x^2) + b \frac{d^2}{dx^2} x + c \frac{d^2}{dx^2} 1</math>

	Now, using the ~~rule for~~ [[differentiation of power functions]], namely <math>d^2(x^n)/dx^2 = n(n-1)x^{n-2}</math>, we obtain:		Now, using the [[differentiation rule for power functions]], namely <math>d^2(x^n)/dx^2 = n(n-1)x^{n-2}</math>, we obtain:

	<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = 2a + 0 + 0 = 2a</math>		<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = 2a + 0 + 0 = 2a</math>

Vipul: /* Second derivative */

2014-05-26T03:11:02Z

Second derivative

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	<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = 2a</math>		<math>\frac{d^2}{dx^2}(ax^2 + bx + c) = 2a</math>

			===Higher derivatives===

			Since the second derivative is a constant (as we find using either of the computational methods above), all higher derivatives are zero.