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Mean value theorem help

  • Thread starter Easty
  • Start date
  • #1
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Homework Statement



If h1 and h2 are contractions on a set B with contraction constants δ1 and δ2 prove that the composite function h2 ° h 1 is also a contraction on B and find a contraction constant.


Homework Equations




|f(a) - f(b)| ≤ δ |a-b|

f '(c) = (f(a)-f(b))/(a-b)


(g°f) '(c) = g '(f(c))x f '(c)


The Attempt at a Solution



So far i'm pretty sure i have to use the mean value theorem and the chain rule. using the mean value on the composite function i get :

|h2(h1(a)) - h2(h1(b))| = |h2 °h1 )' (c)| |a-b|

i get stuck here, i think i should now use the chain rule for the derivaitve term out the front of the equality to somehow make an inequality. am i on the right track?
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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I don't see why you should use derivatives at all. If [itex]h(x)= h_2(h_1(x))[/itex] then [itex]|h(a)- h(b)|= |h_2(h_1(a))- h_2(h_1(b))|\le \delta_2|h_1(a)- h_1(b)|[/itex] and repeat.
 
  • #3
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I don't see why you should use derivatives at all. If [itex]h(x)= h_2(h_1(x))[/itex] then [itex]|h(a)- h(b)|= |h_2(h_1(a))- h_2(h_1(b))|\le \delta_2|h_1(a)- h_1(b)|[/itex] and repeat.

but then arent you assumeing that h(x)= h_2(h_1(x)) is a contration?
 

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