Mathematics - Application of Derivative Question with Solution | TestHub

(i) $f(x)=\sqrt{x}$ is continuous in $[4,9]$ (ii) ${\mathrm{f}}^{\prime }(\mathrm{x})=\frac{1}{2\sqrt{\mathrm{x}}}$ Thus $\mathrm{f}(\mathrm{x})$ is differentiable in $(4,9)$ (iii) $\mathrm{f}(4)\ne \mathrm{f}(9)$. All the three conditions of $\mathrm{LMV}$ theorem satisfied then there exist at least one $\mathrm{c}\in (4,9)$ such that. $$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}\Rightarrow \frac{1}{2\sqrt{c}}=\frac{1}{5}\Rightarrow c=\frac{25}{4}$$