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∈(a,b),c′∈(a′,b′) such that f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a), f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'), and c<c′.
B. Szokefalvi Nagy functioncalculusderivativeanalytic geometrygraphing linesslopereal analysis