Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

For Rolle's theorem in $[\mathrm{a},\mathrm{b}],\mathrm{f}(\mathrm{a})=\mathrm{f}(\mathrm{b}),$ $\mathrm{In}[0,1]\Rightarrow \mathrm{f}(0)=\mathrm{f}(1)=0$ $\because$ the function has to be continuous in [0,1] $\Rightarrow f(0)={\lim }_{x\to 0^{+}}f(x)=0$ $\Rightarrow {\lim }_{x\to 0}{x}^{\alpha }\log x=0\Rightarrow {\lim }_{x\to 0}\frac{\log x}{x^{-\alpha }}=0$ Applying L.H. Rule ${\lim }_{x\to 0}\frac{1/x}{-\alpha x^{-\alpha -1}}=0$ $\Rightarrow {\lim }_{x\to 0}\frac{-x^{\alpha }}{\alpha }=0\Rightarrow \alpha >0$