SCHWEITZER 1975 Problem 7

Let $a<a'<b<b'$ be real numbers and let the real function $f$ be continuous on the interval $[a,b']$ and differentiable in its interior. Prove that there exist $c \in (a,b), c'\in (a',b')$ such that $$f(b)-f(a)=f'(c)(b-a),$$ $$f(b')-f(a')=f'(c')(b'-a'),$$ and $c<c'$ .

B. Szokefalvi Nagy