JEE Advanced Model Paper 5 with solutions for online practice

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)