Twice differentiable

  • Thread starter kathrynag
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  • #1
kathrynag
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Let g:[0,1]-->R be twice differentiable(both g and g' are differentiable functions) with g''(x)>0 for all x in [0,1]. If g(0)>0 and g(1)=1, show that g(d)=d for some point d in (0,1) iff g'(1)>1.


I thought I might use the MVT.
g'(c)=g(1)-g(0)/1=1-g(0)
g'(c)<0 then
 

Answers and Replies

  • #2
36,334
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It might be helpful to sketch a graph of a function that meets these criteria. The graph should be concave up, have a y intercept that is positive, and g(1) = 1.
You have to show that the graph of g crosses the line y = x iff g'(1) > 1.
 
  • #3
kathrynag
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Ok so I assume g'(1)>1
Then g'(1)=lim[(g(x)-g(1)/(1-x)}=lim[(g(x)-1)/(1-x)]>1
 
  • #4
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Assume g'(1) > 1, then show that for some d in (0, 1), g(d) = d.

Then go the other way - g(d) = d ==> g'(1) = 1.

My suggestion of graphing was to help you get a geometric feel for this problem.
 
  • #5
kathrynag
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I guess I get stuck just assuming g'(1)>1

Then g'(1)=lim[(g(x)-g(1)/(1-x)}=lim[(g(x)-1)/(1-x)]>1
 
  • #6
36,334
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Let f(x) = g(x) - x. Can you do something with that?
 

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