Answer:
Option A
Explanation:
Concept lnvolved If f(x) be least degree polynomial having local maximum and local
minimum at $\alpha$ and $\beta$
Then $f'(x)=\lambda (x-\alpha)(x-\beta)$
Sol.Here $p'(x)=\lambda(x-1)(x-3)$
=$\lambda(x^{2}-4x+3)$
Integrating both sides between 1 to 3
$\int_{1}^{3}p'(x) dx=\int_{1}^{3}\lambda(x^{2}-4x+3)dx $
$\Rightarrow$ $(p(x))_{1}^{3}=\lambda\left(\frac{x^{3}}{3}-2x^{2}+3x\right)_{1}^{3}$
$\Rightarrow$ P(3)-p(1)
=$ \lambda \left((9-18+9)-\left(\frac{1}{3}-2+3)\right)\right)$
$2-6=\lambda\left\{\frac{-4}{3}\right\}$
$\therefore$ $p'(x)=3(x-1)(x-3)$
$\therefore$ $p'(0)=9$