Mathematics - Application of Derivative Question with Solution | TestHub

Let $\varphi (x)=\frac{(x-b)(x-c)}{(a-b)(a-c)}f(a)+\frac{(x-c)(x-a)}{(b-c)(b-a)}f(b)+\frac{(x-a)(x-b)}{(c-a)(c-b)}f(c)-f(x)$ Where $\mathrm{a}<\mathrm{c}<\mathrm{b}$ and ${\mathrm{f}}^{11}(\mathrm{x})$ exists at all points in (a,b). Then, there exists a real number $\mu ,\mathrm{a}<\mu <\mathrm{b}$ such that $\frac{\mathrm{f}(\mathrm{a})}{(\mathrm{a}-\mathrm{b})(\mathrm{a}-\mathrm{c})}+\frac{\mathrm{f}(\mathrm{b})}{(\mathrm{b}-\mathrm{c})(\mathrm{b}-\mathrm{a})}+\frac{\mathrm{f}(\mathrm{c})}{(\mathrm{c}-\mathrm{a})(\mathrm{c}-\mathrm{b})}=$