Rolle’s Theorem and the Mean Value Theorem — Cheat sheet
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Core rule
Continuous on [a,b] and differentiable on (a,b) implies some c∈(a,b) with f′(c)=[f(b)−f(a)]/(b−a). Equal endpoint values give Rolle's theorem.
Watch for
Check the entire interval for corners and discontinuities. A zero derivative at one point does not imply constancy.
Mr. Integral