Question

If $p(x)$ 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 $p(0)$ is equal to:

• A. 12
• B. -24
• C. 6
• D. -12

Answer 1

Answer: (D)

Since $p(x)$ has realtive extreme at $x=1\&2$
So $p^{\prime }(x)=0\text{ at }x=1\&2$
$\Rightarrow p^{\prime }(x)=A(x-1)(x-2)$
$\Rightarrow p(x)=\int A\left(x^2-3x+2\right)dx$
$p(x)=A\left(\frac{x^3}{3}-\frac{3x^2}{2}+2x\right)+C\dots$(1)
$P(1)=8$
From (1)
$8=A\left(\frac{1}{3}-\frac{3}{2}+2\right)+C$
$\Rightarrow 8=\frac{5A}{6}+C$
$\Rightarrow 48=5A+6C\dots$(3)
$P(2)=4$
$\Rightarrow 4=A\left(\frac{8}{3}-6+4\right)+C$
$\Rightarrow 4=\frac{2A}{3}+C$
$\Rightarrow 12=2A+3C\dots$(4)
From $3\&4,C=-12$
So $P(0)=C=-12$