Mathematics - Application of Derivative Question with Solution | TestHub

Since $f(x)$ is differentiable in $[0,4]$, using Lagrange's Mean Value Theorem. $f^{\prime }(b)=\frac{f(4)-f(0)}{4},b\in (0,4)$

Now, $\{f(4){\}}^2-\{f(0){\}}^2=\frac{4\{f(4)-f(0)\}}{4}\{f(4)+f(0)\}=4f^{\prime }(b)\{f(4)+f(0)\}$

Also, from Intermediate Mean Value Theorem, $\frac{f(4)+f(0)}{2}=f(a)\text{ for }a\in (0,4)$

Hence, from (2) $[f(4){]}^2-[f(0){]}^2=8f^{\prime }(b)f(a)$