Lagrange Mean Value Theorem (LMVT)

Let f(x) be a real valued function that satisfies the following conditions:

  • (i) f(x) is continuous on the closed interval [a,b]
  • (ii) f(x) is differentiable on the open interval (a,b)

Then there exists at least one point c ∈ (a,b) such that

f′(c) = [f(b) − f(a)]/(b − a)

Some Observations

If f(a) = f(b), then the law of the mean reduces to the Rolle’s theorem.