F(x) is continuous on [a, b], f(a)=0, f(b)= 1, so there exists a certain c∈(a, b) from the intermediate value theorem, so f(c)= 1/2.

By using Lagrange mean value theorem in the interval [a, c] and [c, b] respectively, we can get

ξ∈(a, c) exists, so f' (ξ) = [f (c)-f (a)]/(c-a) =1/2 (c-a).

2(c-a)= 1/f'(ξ)

η∈(c, b) exists such that f' (η) = [f (b)-f (c)]/(b-c) =1/2 (b-c).

2b-c= 1/f'(η)

Add up to get the formula to prove.

Because ξ ∈ (a, c) and┨ ∈ (c, b), ξ and┨ are not equal.