Which set of conditions is required in Rolle's theorem on $[a,b]$?
A$f$ continuous on $(a,b)$, differentiable on $[a,b]$, and $f(a)\ne f(b)$
B$f$ continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$
C$f$ differentiable everywhere and strictly increasing on $[a,b]$
D$f$ continuous on $(a,b)$ only
Answer & Solution
Correct answer: B. $f$ continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$
Rolle's theorem requires three conditions: continuity on the closed interval $[a,b]$, differentiability on the open interval $(a,b)$, and equal endpoint values $f(a)=f(b)$. Then there exists some $c\in(a,b)$ such that $f'(c)=0$.
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