Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Apply Rolle's Theorem, twice, we get required value