Question

Let $P\left(x\right)$ be the polynomial $x^3+a{x}^2+bx+c$, where $a$, $b$, $c$ $\in$ $R$. If $P(-3)=P(+2)=0$ and $P^{\prime }(-3)<0$, which of the following is a possible value of ${}^{\prime }{c}^{\prime }?$

• A. $-27$
• B. $-18$
• C. $-6$
• D. $-3$

Answer 1

Answer: (A)

$-27$

Given, $p\left(x\right)=x^3+a{x}^2+bx+c$
$f^{\prime }\left(x\right)=3x^2+2ax+b$
Since $p\left(x\right)$ posses real solution
$\Rightarrow f^{\prime }\left(x\right)$ should have 2 real roots
$\Rightarrow (2a{)}^2-\mathrm{4.3.}b>0$
$\Rightarrow a^2>3b.......\left(1\right)$
given $p(-3)=-27+9a-3b+c=0$
$\Rightarrow c=27+3b-9a$
for minimum value of $c$, $b$ should be minimum and $a$ should be minimum according to (1)
i.e $a=3,b=1$
Hence $c_{min}=27$