Mathematics - Application of Derivative Question with Solution | TestHub

$f^{\prime }(x)$ is differentiable $\forall x\in [1,6]$ By Lagrange's mean value theorem, $\begin{array}{l}{\mathrm{f}}^{\prime }(\mathrm{x})=\frac{\mathrm{f}(6)-\mathrm{f}(1)}{6-1}\\ {\mathrm{f}}^{\prime }(\mathrm{x})\ge 2\forall \mathrm{x}\in [1,6]\end{array}$ (given) $\Rightarrow \frac{\mathrm{f}(6)+2}{5}\ge 2[\because \mathrm{f}(1)=-2]$ $\Rightarrow \mathrm{f}(6)\ge 10-2\Rightarrow \mathrm{f}(6)\ge 8$