Mean value theorem
Formal statement

The mean value theorem (MVT) can be formally stated as follows. Let f: → R {\displaystyle f:\to \mathbb {R} } be a continuous function on the closed interval {\displaystyle }, and differentiable on the open interval (a, b) {\displaystyle (a,b)}, where a < b {\displaystyle a<b}. Then there exists some c {\displaystyle c} in (a, b) {\displaystyle (a,b)} such that

The mean value theorem is a generalization of Rolle's theorem, which assumes f (a) = f (b) {\displaystyle f(a)=f(b)}, so that the right-hand side above is zero.

The mean value theorem is still valid in a slightly more general setting. One only needs to assume that f: → R {\displaystyle f:\to \mathbb {R} } is continuous on {\displaystyle }, and that for every x {\displaystyle x} in (a, b) {\displaystyle (a,b)} the limit

exists as a finite number or equals ∞ {\displaystyle \infty } or − ∞ {\displaystyle -\infty }. If finite, that limit equals f ′ (x) {\displaystyle f'(x)}. An example where this version of the theorem applies is given by the real-valued cube root function mapping x ↦ x 1 / 3 {\displaystyle x\mapsto x^{1/3}}, whose derivative tends to infinity at the origin.

The theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f (x) = e x i {\displaystyle f(x)=e^{xi}} for all real x {\displaystyle x}. Then

while f ′ (x) ≠ 0 {\displaystyle f'(x)\neq 0} for any real x {\displaystyle x}.

These formal statements are also known as Lagrange 's Mean Value Theorem.
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