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| {{Calculus}}
| | I am 24 years old and my name is Brigette Tait. I life in Cruzeiro (Brazil).<br><br>My web site ... [http://ruthculloch75.blogspot.com/ diet plans] |
| [[File:RTCalc.svg|thumb|300 px|right|If a [[real number|real]]-valued function ''ƒ'' is [[continuous function|continuous]] on a [[closed interval]] [''a'', ''b''], [[derivative|differentiable]] on the [[open interval]] (''a'', ''b''), and ''ƒ''(''a'') = ''ƒ''(''b''), then there exists a ''c'' in the open interval (''a'', ''b'') such that <math>f'(c) = 0.</math>]]
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| In [[calculus]], '''Rolle's theorem''' essentially states that any real-valued [[differentiable function]] that attains equal values at two distinct points must have a [[stationary point|point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero]].
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| == Standard version of the theorem ==
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| If a [[real number|real]]-valued function ''f'' is [[continuous function|continuous]] on a [[closed interval]] [''a'', ''b''], [[derivative|differentiable]] on the [[open interval]] (''a'', ''b''), and ''f''(''a'') = ''f''(''b''), then there exists a ''c'' in the open interval (''a'', ''b'') such that
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| :<math>f'(c) = 0</math>.
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| This version of Rolle's theorem is used to prove the [[mean value theorem]], of which Rolle's theorem is indeed a special case. It is also the basis for the proof of [[Taylor's theorem]].
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| == History ==
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| Indian mathematician [[Bhāskara II]] (1114–1185) is credited with knowledge of Rolle's theorem.<ref>R.C. Gupta, ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Westen Cultures'', [http://books.google.ca/books?id=raKRY3KQspsC&pg=PA156 p. 156].</ref>
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| The first known formal proof was offered by [[Michel Rolle]] in 1691 and used the methods of differential calculus.
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| The name "Rolle's theorem" was first used by [[Moritz Wilhelm Drobisch]] of Germany in 1834 and by [[Giusto Bellavitis]] of Italy in 1846.<ref>See [[Florian Cajori]]'s ''A History of Mathematics'', p. 224 [http://books.google.com/books?id=mGJRjIC9fZgC&pg=RA1-PA119-IA5&lpg=RA1-PA119-IA5&dq=Giusto+Bellavitis++rolle&source=bl&ots=cN8cBV6rJ1&sig=scHAQEZt_PzsfigLvDFJ3Jz6_dU&hl=en&sa=X&oi=book_result&resnum=2&ct=result].</ref>
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| == Examples ==
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| ===First example===
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| [[File:semicircle.svg|thumb|300px|A '''semicircle''' of radius ''r''.]]
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| For a radius ''r'' > 0, consider the function
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| <math>f(x)=\sqrt{r^2-x^2},\quad x\in[-r,r].</math>
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| Its [[graph of a function|graph]] is the upper [[semicircle]] centered at the origin. This function is continuous on the closed interval [−''r'',''r''] and differentiable in the open interval (−''r'',''r''), but not differentiable at the endpoints −''r'' and ''r''. Since ''f''(−''r'') = ''f''(''r''), Rolle's theorem applies, and indeed, there is a point where the derivative of ''f'' is zero. Note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
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| <br style="clear:both;"><!--This break ensures that the text of the next section is clear of the image in the current section--> | |
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| ===Second example===
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| [[File:Absolute value.svg|thumb|300px|The graph of the absolute value function.]]
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| If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the [[absolute value]] function
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| :<math>f(x) = |x|,\qquad x\in[-1,1].</math>
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| Then ''f''(−1) = ''f''(1), but there is no ''c'' between −1 and 1 for which the derivative is zero. This is because that function, although continuous, is not differentiable at ''x'' = 0. Note that the derivative of ''f'' changes its sign at ''x'' = 0, but without attaining the value 0. The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, ''f'' will still have a critical number in the open interval (a,b), but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
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| <br style="clear:both;"><!--This break ensures that the text of the next section is clear of the image in the current section--> | |
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| == Generalization ==
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| The second example illustrates the following generalization of Rolle's theorem:
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| Consider a real-valued, continuous function ''f'' on a closed interval [''a'',''b''] with ''f''(''a'') = ''f''(''b''). If for every ''x'' in the open interval (''a'',''b'') the right-hand limit
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| :<math>f'(x+):=\lim_{h \to 0^+}\frac{f(x+h)-f(x)}{h}</math>
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| and the left-hand limit
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| :<math>f'(x-):=\lim_{h \to 0^-}\frac{f(x+h)-f(x)}{h}</math>
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| exist in the [[extended real line]] [−∞,∞], then there is some number ''c'' in the open interval (''a'',''b'') such that one of the two limits
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| :<math>f'(c+)\quad\text{and}\quad f'(c-)</math>
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| is ≥ 0 and the other one is ≤ 0 (in the extended real line). If the right- and left-hand limit agree for every ''x'', then they agree in particular for ''c'', hence the derivative of ''f'' exists at ''c'' and is equal to zero.
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| ===Remarks===
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| #If ''f'' is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers.
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| #This generalized version of the theorem is sufficient to prove [[Convex function|convexity]] when the one-sided derivatives are [[monotonically increasing]]:<ref>{{citation |last=Artin |first=Emil |authorlink=Emil Artin |others=trans. Michael Butler |title=The Gamma Function |origyear=1931 |year=1964 |publisher=[[Henry Holt and Company|Holt, Rinehart and Winston]] |pages=3–4}}</ref>
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| ::<math>f'(x-) \le f'(x+) \le f'(y-),\qquad x < y.</math>
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| == Proof of the generalized version ==
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| Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization.
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| The idea of the proof is to argue that if ''f''(''a'') = ''f''(''b''), then ''f'' must attain either a [[maxima and minima|maximum]] or a minimum somewhere between ''a'' and ''b'', say at ''c'', and the function must change from increasing to decreasing (or the other way around) at ''c''. In particular, if the derivative exists, it must be zero at ''c''.
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| By assumption, ''f'' is continuous on [''a'',''b''], and by the [[extreme value theorem]] attains both its maximum and its minimum in [''a'',''b'']. If these are both attained at the endpoints of [''a'',''b''], then ''f'' is [[constant function|constant]] on [''a'',''b''] and so the derivative of ''f'' is zero at every point in (''a'',''b'').
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| Suppose then that the maximum is obtained at an [[interior point]] ''c'' of (''a'',''b'') (the argument for the minimum is very similar, just consider −''f ''). We shall examine the above right- and left-hand limits separately.
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| For a real ''h'' such that ''c'' + ''h'' is in [''a'',''b''], the value ''f''(''c'' + ''h'') is smaller or equal to ''f''(''c'') because ''f'' attains its maximum at ''c''. Therefore, for every ''h'' > 0,
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| :<math>\frac{f(c+h)-f(c)}{h}\le0,</math>
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| hence
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| :<math>f'(c+):=\lim_{h \to 0^+}\frac{f(c+h)-f(c)}{h}\le0,</math>
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| where the limit exists by assumption, it may be minus infinity.
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| Similarly, for every ''h'' < 0, the inequality turns around because the denominator is now negative and we get
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| :<math>\frac{f(c+h)-f(c)}{h}\ge0,</math>
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| hence
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| :<math>f'(c-):=\lim_{h \to 0^-}\frac{f(c+h)-f(c)}{h}\ge0,</math>
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| where the limit might be plus infinity.
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| Finally, when the above right- and left-hand limits agree (in particular when ''f'' is differentiable), then the derivative of ''f'' at ''c'' must be zero.
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| == Generalization to higher derivatives ==
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| We can also generalize Rolle's theorem by requiring that ''f'' has more points with equal values and greater regularity. Specifically, suppose that
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| * the function ''f'' is ''n'' − 1 times continuously differentiable on the closed interval [''a'',''b''] and the ''n''<sup>th</sup> derivative exists on the open interval (''a'',''b''), and
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| * there are ''n'' intervals given by ''a''<sub>1</sub> < ''b''<sub>1</sub> ≤ ''a''<sub>2</sub> < ''b''<sub>2</sub> ≤ . . .≤ ''a<sub>n</sub>'' < ''b<sub>n</sub>'' in [''a'',''b''] such that ''f''(''a<sub>k</sub>'') = ''f''(''b<sub>k</sub>'') for every ''k'' from 1 to ''n''.
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| Then there is a number ''c'' in (''a'',''b'') such that the ''n''<sup>th</sup> derivative of ''f'' at ''c'' is zero.<!-- boy, a diagram sure would be handy here to explain this; just even showing a function with three roots and its derivative with one root would do-->
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| <!--Urgh: Of course, t-->The requirements concerning the ''n''<sup>th</sup> derivative of ''f'' can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with ''f''<sup> (''n''−1)</sup> in place of ''f''.
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| ===Proof===
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| The proof uses [[mathematical induction]]. For ''n'' = 1 is simply the standard version of Rolle's theorem. As induction hypothesis, assume the generalization is true for ''n'' − 1. We want to prove it for ''n'' > 1. By the standard version of Rolle's theorem, for every integer ''k'' from 1 to ''n'', there exists a ''c<sub>k</sub>'' in the open interval (''a<sub>k</sub>'',''b<sub>k</sub>'') such that ''f<nowiki>'</nowiki> ''(''c<sub>k</sub>'') = 0. Hence the first derivative satisfies the assumptions with the ''n'' − 1 closed intervals [''c''<sub>1</sub>,''c''<sub>2</sub>], . . ., [''c''<sub>''n''−1</sub>,''c<sub>n</sub>'']. By the induction hypothesis, there is a ''c'' such that the (''n'' − 1)<sup>st</sup> derivative of ''f<nowiki>'</nowiki> '' at ''c'' is zero.
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| == Generalizations to other fields ==
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| Rolle's theorem is a property of differentiable functions over the real numbers, which are an [[ordered field]]. As such, it does not generalize to other [[field (mathematics)|fields]], but the following corollary does: if a real polynomial splits (has all its roots) over the real numbers, then its derivative does as well – one may call this property of a field '''Rolle's property'''. More general fields do not always have a notion of differentiable function, but they do have a notion of polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field.
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| Thus Rolle's theorem shows that the real numbers have Rolle's property, and any algebraically closed field such as the [[complex numbers]] has Rolle's property, but conversely the rational numbers do not – for example, <math>x^3-x=x(x-1)(x+1)</math> splits over the rationals, but its derivative <math>3x^2-1 = 3(x-1/\sqrt{3})(x+1/\sqrt{3})</math> does not. The question of which fields satisfy Rolle's property was raised in {{Harv|Kaplansky|1972}}. For [[finite field]]s, the answer is that only <math>\mathbf{F}_2</math> and <math>\mathbf{F}_4</math> have Rolle's property; this was first proven via technical means in {{Harv|Craven|Csordas|1977}}, and a simple proof is given in {{Harv|Ballantine|Roberts|2002}}.
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| For a complex version, see [[Voorhoeve index]].
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| == See also ==
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| *[[Mean value theorem]]
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| *[[Intermediate value theorem]]
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| *[[Linear interpolation]]
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| *[[Gauss-Lucas theorem]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| {{refbegin}}
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| * {{ Citation | first = Irving | last = Kaplansky | authorlink = Irving Kaplansky | title = Fields and Rings | year = 1972 }}
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| * {{ Citation | title = Multiplier sequences for fields | first1 = Thomas | last1 = Craven | first2 = George | last2 = Csordas | journal = Illinois J. Math. | volume = 21 | year = 1977 | pages = 801–817 | url = http://projecteuclid.org/euclid.ijm/1256048929 | issue = 4 }}
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| * {{ Citation | title = A Simple Proof of Rolle's Theorem for Finite Fields | first1 = C. | last1 = Ballantine | first2 = J. | last2 = Roberts | journal = [[The American Mathematical Monthly]] | volume = 109 |date=January 2002 | pages = 72–74 | issue = 1 | doi = 10.2307/2695770 | jstor = 2695770 | publisher = Mathematical Association of America }}
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| {{refend}}
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| == External links ==
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| * {{springer|title=Rolle theorem|id=p/r082550}}
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| * [http://www.cut-the-knot.org/Curriculum/Calculus/MVT.shtml Rolle's and Mean Value Theorems] at [[cut-the-knot]].
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| {{DEFAULTSORT:Rolle's Theorem}}
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| [[Category:Theorems in real analysis]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in calculus]]
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