Mean value theorem (divided differences)

<h2 id="statement-of-the-theorem">Statement of the theorem</h2>
For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point

ξ
        ∈
        (
        min
        {
        
          x
          
            0
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        }
        ,
        max
        {
        
          x
          
            0
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        }
        )
        
      
    
    {\displaystyle \xi \in (\min\{x_{0},\dots ,x_{n}\},\max\{x_{0},\dots ,x_{n}\})\,}

where the nth derivative of f equals n ! times the nth <a href="/facts/Divided_difference/HUIoAusz">divided difference</a> at these points:

f
        [
        
          x
          
            0
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ]
        =
        
          
            
              
                f
                
                  (
                  n
                  )
                
              
              (
              ξ
              )
            
            
              n
              !
            
          
        
        .
      
    
    {\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.}

For n = 1, that is two function points, one obtains the simple <a href="/facts/Mean_value_theorem/JnjkzQb4">mean value theorem</a>.

<h2 id="proof">Proof</h2>
Let 
 
 
 
 P
 
 
 {\displaystyle P}
 
 be the <a href="/facts/Lagrange_interpolation_polynomial/MnpMmhtC">Lagrange interpolation polynomial</a> for f at x0, ..., xn.
Then it follows from the <a href="/facts/Newton_polynomial/2vRsjpA2">Newton form</a> of 
 
 
 
 P
 
 
 {\displaystyle P}
 
 that the highest order term of 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is 
 
 
 
 f
 [
 
 x
 
 0
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 ]
 
 x
 
 n
 
 
 
 
 {\displaystyle f[x_{0},\dots ,x_{n}]x^{n}}
 
.
Let 
 
 
 
 g
 
 
 {\displaystyle g}
 
 be the remainder of the interpolation, defined by 
 
 
 
 g
 =
 f
 −
 P
 
 
 {\displaystyle g=f-P}
 
. Then 
 
 
 
 g
 
 
 {\displaystyle g}
 
 has 
 
 
 
 n
 +
 1
 
 
 {\displaystyle n+1}
 
 zeros: x0, ..., xn.
By applying <a href="/facts/Rolle%2527s_theorem/M8oMH2gp">Rolle's theorem</a> first to 
 
 
 
 g
 
 
 {\displaystyle g}
 
, then to 
 
 
 
 
 g
 ′
 
 
 
 {\displaystyle g'}
 
, and so on until 
 
 
 
 
 g
 
 (
 n
 −
 1
 )
 
 
 
 
 {\displaystyle g^{(n-1)}}
 
, we find that 
 
 
 
 
 g
 
 (
 n
 )
 
 
 
 
 {\displaystyle g^{(n)}}
 
 has a zero 
 
 
 
 ξ
 
 
 {\displaystyle \xi }
 
. This means that

0
        =
        
          g
          
            (
            n
            )
          
        
        (
        ξ
        )
        =
        
          f
          
            (
            n
            )
          
        
        (
        ξ
        )
        −
        f
        [
        
          x
          
            0
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ]
        n
        !
      
    
    {\displaystyle 0=g^{(n)}(\xi )=f^{(n)}(\xi )-f[x_{0},\dots ,x_{n}]n!}
  
,

<h2 id="applications">Applications</h2>
The theorem can be used to generalise the <a href="/facts/Stolarsky_mean/TcUUbWGR">Stolarsky mean</a> to more than two variables.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">de Boor, C. (2005). "Divided differences". Surv. Approx. Theory. 1: 46–69. MR 2221566. <a href="/wiki/Carl_R._de_Boor" target="_blank">/wiki/Carl_R._de_Boor</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Mean value theorem (divided differences) open-in-new

Mean value theorem (divided differences)