Mathematics - Application of Derivative Question with Solution | TestHub

By LMVT, $\exists a\in (0,4)\ni \frac{f(4)-f(0)}{4-0}=f^{{}^{\prime }}(a)\Rightarrow f(4)-f(0)=4f^{{}^{\prime }}(a)$

$\because \frac{\mathrm{f}(4)+\mathrm{f}(0)}{2}$ lies between $\mathrm{f}(0)$ and $\mathrm{f}(4)$, by Intermediate value theorem $\exists b\in (0,4)\ni \frac{f(4)+f(0)}{2}=f(b)$ hence, $\left(f(4{)}^2\right)-(f(0){)}^2=8f^{{}^{\prime }}(a)f(b)$