Mean Value Theorem - Formal Statement

Formal Statement

Topics in calculus
  • Fundamental theorem
  • Limits of functions
  • Continuity
  • Mean value theorem
  • Rolle's theorem
Differential calculus
  • Derivative
  • Second derivative
  • Third derivative
  • Change of variables
  • Implicit differentiation
  • Taylor's theorem
  • Related rates
  • Rules and identities
    Power rule
    Product rule
    Quotient rule
    Sum rule
    Chain rule
Integral calculus
  • Lists of integrals
  • Improper integral
  • Multiple integral
  • Integration by
    parts
    disks
    cylindrical shells
    substitution
    trigonometric substitution
    partial fractions
    changing order
Vector calculus
  • Gradient
  • Divergence
  • Curl
  • Laplacian
  • Gradient theorem
  • Green's theorem
  • Stokes' theorem
  • Divergence theorem
Multivariable calculus
  • Matrix calculus
  • Partial derivative
  • Multiple integral
  • Line integral
  • Surface integral
  • Volume integral
  • Jacobian

Let f : → R be a continuous function on the closed interval, and differentiable on the open interval (a, b), where a < b. Then there exists some c in (a, b) such that

The mean value theorem is a generalization of Rolle's theorem, which assumes 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 is continuous on, and that for every x in (a, b) the limit

exists as a finite number or equals +∞ or −∞. If finite, that limit equals f′(x). An example where this version of the theorem applies is given by the real-valued cube root function mapping x to x1/3, whose derivative tends to infinity at the origin.

Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f(x) = eix for all real x. Then

f(2π) − f(0) = 0 = 0(2π − 0)

while |f′(x)| = 1.

Read more about this topic:  Mean Value Theorem

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