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Flett’s mean value theorem - mklabgr - 06-22-2026
Flett’s mean value theorem
Summary
Flett’s Mean Value Theorem is a refinement of the classical Mean Value Theorem proved by Thomas M. Flett in 1958. It states that if a function (f) is differentiable on ($[a,b]$) and its derivatives at the endpoints are equal, ($f'(a)=f'(b)$), then there exists a point $(c\in(a,b))$ such that $(f©-f(a)=(c-a)f'©)$. Geometrically, this means that the tangent line at © passes through the point ((a,f(a))), providing a different kind of “mean value” relationship than the classical theorem.
The article presents an elegant proof based on Rolle’s Theorem, discusses cases where multiple such points © may exist, illustrates the theorem with the function (\sin x), and concludes with applications to integral equations, highlighting the theorem’s usefulness and its role in inspiring several later generalizations in analysis.
ARTICLE