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Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Suppose that f(a) = a and f(b) = b. Show that there is c ∈ (a, b) such that f′(c) = 1. Further, show that there are distinct c1, c2 ∈ (a, b) such that f′(c1) + f′(c2) = 2.

Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Suppose that f(a) = a
and f(b) = b. Show that there is c ∈ (a, b) such that f′(c) = 1. Further, show that there
are distinct c1, c2 ∈ (a, b) such that f′(c1) + f′(c2) = 2.

Grade:12

4 Answers

Anoopam Mishra
126 Points
5 years ago
Using Lagrange mean value theorem we know that f`(c)=(f(b)-f(a))\div (b-a) for atleast one c\in \left [ a,b \right ]. You can see easily that f`(c)=1.
Now the second part, take y = (a+b)/2. We will apply LMVT for two intervals \left [ a,a+b/2 \right ] and \left [ a+b/2,b \right ]. There exists atleast one c1\in \left [ a,a+b/2 \right ] and\ c2\in \left [ a+b/2,b \right ] for which f`(c1)=f(y)-f(a)/(y-a) = f(y)-a/(a+b)/2-a = 2f(y)-2a/b-a\ and\ f`(c2)=f(b)-f(y)/b-y = b-f(y)/b-(a+b)/2 = 2b-2f(y)/b-a.
It is easy to see that f`(c1)+f`(c2) = 2.
Aakash
110 Points
5 years ago
BY LMVT(Lagrange mean value theorem)
Apply theorem in at points (a,f(a)) & (b,f(b))
 
\frac{f(a)-f(b)}{b-a} = f`(c)  a<c<b
\frac{b-a}{b-a} =1 = f`(c) This prove the first part that there exist a point c between a & b such that f`(c) = 1
 
SECOND PART
 
now break the interval (a,b) in two Parts in ( a, \right\frac{a+b}{2} ) &\left (\frac{a+b}{2}, \right b )
 
now apply LMVT indivisually in both interval
 
\frac{f(\frac{a+b}{2}) -f(a)}{\frac{a+b}{2}-a} =f`(c_{1})  ...............................(1) a<c_{1}<\frac{a+b}{2}
\frac{f(b)-f(\frac{a+b}{2}) }{b-\frac{a+b}{2}} =f`(c_{2}) ...............................(2)\frac{a+b}{2}<c_{2}<b
Adding (1) and (2)
also put f(a) =a And f(b)=b
you got
 
f`(c_{1}) + f`(c_{2}) =2
 
This proves second part also that there exist c1 & c2 between (a,b) such that f’(c1)+f’(c2) =2
 
PLEASE APPROVE
Anoopam Mishra
126 Points
5 years ago
My answer was exactly the same as of Akash. But you disapproved my answer. 
 
Anoopam Mishra
126 Points
5 years ago
This is not fair!!!!!

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