Let f(x)=αx^2+βx+γ be a quadratic function, so α≠0, and let I=[a,b].

Question

a) Check f satisfies the hypothesis of the Mean Value Theorem.

b)Show that the number c ∈ (a,b) in the Mean Value Theorem is the midpoint of the interval I.

Answer 1

every polynomial is continuous and differentiable everywhere.

At the midpoint, x = (a+b)/2

f'(x) = 2αx+β
So, show that f'((a+b)/2) = (f(b)-f(a)/(b-a)