Discussion Forum

Let p(x) be a  be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x=3. If p(1)=6 and P(3)=2 , the p '(0) is 

Answer:

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)

    =$\lambda \left((9-18+9)-\left(\frac{1}{3}-2+3)\right)\right)$

$2-6=\lambda\left\{\frac{-4}{3}\right\}$

$\therefore$  $p'(x)=3(x-1)(x-3)$

$\therefore$  $p'(0)=9$