proof of extended mean-value theorem
Let f:[a,b]→ℝnormal-:fnormal-→abℝf:[a,b]\to\mathbb{R} and g:[a,b]→ℝnormal-:gnormal-→abℝg:[a,b]\to\mathbb{R} be continuous on [a,b]ab[a,b] and differentiable on (a,b)ab(a,b). Define the function
Because fff and ggg are continuous on [a,b]ab[a,b] and differentiable on (a,b)ab(a,b), so is hhh. Furthermore, h(a)=h(b)=0hahb0h(a)=h(b)=0, so by Rolle’s theorem there exists a ξ∈(a,b)ξab\xi\in(a,b) such that h′(ξ)=0superscripthnormal-′ξ0h^{{\prime}}(\xi)=0. This implies that
and, if g(b)≠g(a)gbgag(b)\neq g(a),
