Mathematics - Application of Derivative Question with Solution | TestHub

Consider the function $\mathrm{f}$ defined by $\mathrm{f}(\mathrm{x})={\mathrm{a}}_0\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}+{\mathrm{a}}_{\mathrm{n}}\frac{{\mathrm{x}}^{\mathrm{n}}}{\mathrm{n}}+\dots +{\mathrm{a}}_{\mathrm{n}-1}\frac{{\mathrm{x}}^2}{2}+{\mathrm{a}}_{\mathrm{n}}\mathrm{x}$ Since, $f(x)$ is a polynomial, so it is continuous and differentiable for all $\mathrm{x}$. $f(x)$ is continuous in the closed interval $[0,1]$ and differentiable in the open interval $(0,1)$. Also, $\mathrm{f}(0)=0$ and $f(1)=\frac{a_0}{n+1}+\frac{a_1}{n}+\dots .+\frac{a_{n-1}}{2}+a_n=0[\text{ say }]$ i.e. $f(0)=f(1)$ Thus, all the three conditions of Rolle's theorem are satisfied. Hence, there is atleast one value of $x$ in the open interval $(0,1)$ where $f^{\prime }(x)=0$ i.e. $a_0{x}^n+a_1{x}^{n-1}+\dots .+a_n=0$