Let 𝑓(𝑥) = 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 . Prove that for any interval [𝑎 , 𝑏] the value 𝑐 guaranteed by the mean value theorem is the midpoint of the interval.

Answer 1

f(x) = px² + qx + r

f'(x) = 2px + q

f'(c) = [f(b) - f(a)] / (b - a) = [(pb² + qb + r) - (pa² + qa +r)] / (b-a)

2pc + q = [p(b² - a²) + q(b-a)] / (b - a)

2pc + q = [p(b - a)(b + a) + q(b - a)] / (b - a)

2pc + q = p(b + a) + q

2pc = p(b + a)

c = (b + a) / 2