Mean Value Theorem | Rolle's Theorem | Lagrange Theorem | Cauchy's Mean Value Theorem

Rolle's Theorem: If f(x) is a function defined in an interval [a, b] such that 
I. f(x) is continous in [a, b]
II. f'(x) exists in (a, b)
III. f(a) = f(b) then there exists c ∈ (a, b) such that f'(c) = 0 

Lagrange Theorem:  If f(x) is a function defined in an interval [a, b] such that 
I. f(x) is continous in [a, b]
II. f'(x) exists in (a, b) then there exists c ∈ (a, b) such that f'(c) = \({f(b) - f(a) \over b-a}\)

Cauchy's Mean Value Theorem: If f(x) and g(x) are two functions defined into interval [a, b] such that
I. f(x) and g(x) are continous in [a, b]
II. f'(x) and g'(x) exist in (a, b)
III. g'(x) ≠ 0, ∀x ∈ (a, b) then there exists c ∈ (a, b) such that \( {f'(c) \over g'(c)} = {f(b) - f(a) \over g(b) - g(a)}\)

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