Lagrange’s Mean value Theorem (LMVT)

Lagrange’s Mean value Theorem (LMVT)

Let f be a function continuous in [a, b] and differentiable in (a, b), then there exists

c ∈ (a, b) such that .

Note that is the slope of the chord AB, where A ≡ (a, f(a)) and B ≡ f(b, f(b)). Hence LMVT asserts that there exists an interior point of [a, b], where tangent is parallel to the chord AB.

For example let f(x) = ln x, x ∈  [1, e]. Obviously f is continuous in[1, e] and differentiable in (1, e). Hence LMVT asserts that there exists c ∈ (1, e) such that

f ′(c) =       ⇒ c = e – 1 ∈ [1, e].

Rolle’s Theorem

Let f be a function continuous in [a, b] and differentiable in (a, b) such that f(a) = f(b), then there exists c ∈ (a, b) such that f ′(c) = 0.

For example let f(x) =  ln (sinx), x ∈ [π/6, 5π/6]. Obviously f is continuous in
[π/6, 5π/6] and differentiable in (π/6, 5π/6). Further f(π/6) = f(5π/6). Hence Rolle’s theorem asserts that there exists c ∈ (π/6, 5π/6) such that f′(c) = 0 ⇒ cotc = 0.

Infact c = π/2∈ (π/6, 5π/6).