# Difference between revisions of "Mean Value Theorem"

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− | The | + | The Mean Value Theorem (often abbreviated MVT) is considered one of the most important results in [[real analysis]]. An elegant proof of the [[Fundamental Theorem of Calculus]] can be given using Mean Value Theorem. |

− | <cmath>f(c)=\ | + | == Statement == |

+ | Let <math>f:[a,b]\rightarrow\mathbb R</math> be differentiable function on the [[interval]] <math>(a,b)</math>. The '''Mean Value Theorem''' states there exists a point <math>c\in(a,b)</math> such that <cmath>f'(c)=\frac{f(b)-f(a)}{b-a}.</cmath> | ||

− | + | == Proof == | |

− | + | Note that <math>\frac{f(b)-f(a)}{b-a}</math> is the slope of the [[secant]] [[line]] which passes through the graph of <math>f</math> at points <math>(a,f(a))</math> and <math>(b,f(b))</math>. Let <math>g(x)=f(x)-\frac{f(b)-f(a)}{b-a}(x-a)</math>. As <math>g</math> is continuous on <math>[a,b]</math> since it is the linear combinatoin of the continuous functions <math>f(x)</math> and <math>(x-a)</math>, and that <cmath>g'(x)=f'(x)-\frac{f(b)-f(a)}{b-a}.</cmath> Since <math>g</math> is differentiable on <math>(a,b)</math> and <math>g(a)=g(b)=f(a)</math>, by [[Rolle's Theorem]], there exists some <math>c\in(a,b)</math> such that <math>g'(c)=0</math>. Thus <cmath>f'(c)=g'(c)+\frac{f(b)-f(a)}{b-a}=\frac{f(b)-f(a)}{b-a}</cmath> as desired. <math>\fbox{}</math> | |

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## Latest revision as of 18:19, 18 September 2020

The Mean Value Theorem (often abbreviated MVT) is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using Mean Value Theorem.

## Statement

Let be differentiable function on the interval . The **Mean Value Theorem** states there exists a point such that

## Proof

Note that is the slope of the secant line which passes through the graph of at points and . Let . As is continuous on since it is the linear combinatoin of the continuous functions and , and that Since is differentiable on and , by Rolle's Theorem, there exists some such that . Thus as desired.

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