I tried the Question as so:
$$f^2(4)-f^2(0)=[f(4)+f(0)]\cdot[f(4)-f(0)]$$
Now $$\frac{f(4)-f(0)}{4-0}= f'(c) ~, ~~c \in (0,4)$$ $$f(4)-f(0)=4f'(c)$$
And
$$\frac{f(4)+f(0)}{2}= f(b) ~~, ~b \in ~ (0,4) $$
$$\therefore f^2(4)-f^2(0) = 4f'(c)\cdot2f(b) $$ $$= 8f'(c)f(b)$$
But the answer in the book says that b and c are the same points and the answer is given as $8f'(a)f(a)$
so basically I need to prove that c and b are the same points.
BTW -> what does this mean $f : [0,4] \rightarrow~~\mathbb{R}$