Answer (8) By LMVT, there exist a ∊ (0, 4) such that
f(4)−f(0)
4−0
= f'(a) ∴ f(4) - f(0) = 4f'(a) ... (i) ∴
f(4)+f(0)
2
lies between f(0) and f(4) by the intermediate value theorem of continuous function,there exist b ∊ (0, 4) such that f(4) + f(0) = 2f(b)...(ii) From equation (i) and (ii) , (f(4))2 - (f(0))2 = 8f'(a) f(b)