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