let F be continuous on [a,b] and differentiable on(a,b). if f(a) = a and f(b) = b,

show that there exixt distinct  c₁ , c₂  in (a,b) such that f '(c₁) + f '(c₂) = 2

 From Lagrange's Mean Value Theorem,

there exist some cE(a,b) such that f'(c)=[f(b)-f(a)]/[b-a]  

here

f'(c)=1

now if f'(c) is atleast a quadratic expression with D>0(unequal roots)

Then there exist two c's or c1 and c2 for which f'(x)=1

That is,

f'(c1)+f'(c2)=2

In this problem you cannot assume anything about f'(x) least of all that it is a quadratic. f(x) need not be a polynomial even

Consider the intervals  and . There exist c₁ and c₂ in these intervals such that

     and

 Thank you sir for pointing at my mistake:)