Question

If $p\left(x\right)$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$; then $p\left(0\right)$ is equal to :

• A. $12$
• B. $-12$
• C. $-24$
• D. $6$

Answer 1

The correct option is B $-12$

$p^{\prime }\left(1\right)=0\&p\left(2\right)=0$

$p^{\prime }\left(x\right)=a(x-1)(x-2)$

$p\left(x\right)=a(\frac{x^3}{3}-\frac{3x^2}{2}+2x)+b$

$p\left(1\right)=8\Rightarrow a(\frac{1}{3}-\frac{3}{2}+2)+b=8\cdots \left(i\right)$

$p\left(2\right)=4.....\left(ii\right)$
from equation $\left(i\right)$ and $\left(ii\right)$

$a=24\&b=-12$

$p\left(0\right)$ $=b=-12$