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Rolle's Theorem

If a continuous curve starts and ends at the same height and has a derivative at every interior point, then somewhere between the endpoints its tangent must be horizontal. Rolle's theorem makes this geometric observation precise.
ROLLE'S THEOREM
Let a<ba<b, and let f:[a,b]→Rf:[a,b]\to\mathbb{R} be continuous on [a,b][a,b] and differentiable on (a,b)(a,b). Suppose that
f(a)=f(b).\begin{equation*}f(a)=f(b).\end{equation*}
Then there is at least one point c∈(a,b)c\in(a,b) such that
f′(c)=0.\begin{equation*}f'(c)=0.\end{equation*}

Proof

Suppose, for a contradiction, that f′(x)≠0f'(x)\ne 0 for every x∈(a,b)x\in(a,b). Since ff is continuous on the closed interval [a,b][a,b], the extreme value theorem ensures that it attains both its absolute minimum mm and its absolute maximum MM on this interval.
Neither extreme value can be attained at an interior point. It follows that both the absolute minimum and the absolute maximum must occur at the endpoints. But the endpoint values are equal, so
m=f(a)=f(b)=M.\begin{equation*}m=f(a)=f(b)=M.\end{equation*}
For every x∈[a,b]x\in[a,b], we have m≤f(x)≤Mm\leq f(x)\leq M. Hence ff is constant on [a,b][a,b], and its derivative is zero throughout (a,b)(a,b). This is again a contradiction. Therefore at least one c∈(a,b)c\in(a,b) satisfies f′(c)=0f'(c)=0.

References

  • [Apostol1967]Apostol, Tom M. Calculus, Volume I: One-Variable Calculus, with an Introduction to Linear Algebra. Second edition. John Wiley \& Sons, 1967.