# Can we write Rolle's Theorem this way?

by nil1996
Tags: change, rolle, theorem, write
 P: 296 Rolle's theorem: Statements: If y =f(x) is a real valued function of a real variable such that: 1) f(x) is continuous on [a,b] 2) f(x) is differentiable on (a,b) 3) f(a) = f(b) then there exists a real number c$\in$(a,b) such that f'(c)=0 what if the the f(x) is like the following graph: here there is a point 'c' for which f'(c) =0 but f(a) $\neq$ f(b) So to take such cases in consideration can we make a change to the last statement of Rolle's theorem as: 3)f(c) > [f(a),f(b)] Or f(c)<[f(a),f(b)] are there any exceptions to the above statement? Attached Thumbnails
 Mentor P: 4,499 Yes, that statement works. The easiest way to prove this is to notice that if f(c.) > f(b) > f(a) (for example), then there exists some d such that a
P: 296
 Quote by Office_Shredder Then applying Rolle's theorem to the interval [d,c] completes the proof.
Do you mean interval[d,b]

Mentor
P: 4,499

## Can we write Rolle's Theorem this way?

Yeah good catch
 P: 296 Thanks

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