$\text{ If the function }f:[0,4]\to R\text{ is differentiable then }a,b\in (0,4)\frac{f^2\left(4\right)-f^2\left(0\right)}{f\left(a\right)\cdot f\left(b\right)}=$

$\text{ since }f\text{ is differentiable on }[0,4]\text{ so it is continuous also. Hence by LMTV }$

$\begin{array}{r}\mathrm{\exists }\text{ Some }\mathrm{a}\in (0,4)\text{ such that }\\ f^{\mathrm{\prime }}\left(a\right)=\frac{f\left(4\right)-f\left(0\right)}{4-0}----\left(1\right)\end{array}$

Again, by intermediate value theorem for continuous functions,    

$\mathrm{\exists }\mathrm{b}\in (0,4)\text{ such that }f\left(b\right)=\frac{f\left(0\right)+f\left(4\right)}{2}----\left(2\right)$

From (1) and (2)

$f^{\mathrm{\prime }}\left(a\right)\cdot f^{\mathrm{\prime }}\left(b\right)=\frac{f^2\left(4\right)-f^2\left(0\right)}{8}$