Question

The polynomial $P ( x )= x ^3+ ax ^2+ bx + c$ has the property that the mean of its roots, the product of its roots and the sum of its coefficient are all equal. If the $y$-intercept of the graph of $y=P(x)$ is 2 , then 4
Minimum value of $c x^2+2 a x-b$ is equal to

• A. - 7
• B. - 1
• C. 2
• D. 7

Answer 1

Answer: (A)

$\because$ y-intercept $=c=2$
Mean of its root $=$ product of roots
$\Rightarrow \frac{-a}{3}=-c \Rightarrow a=6 $
$\because 1+a+b+c=\text { sum of coefficient }=-2$
$\therefore b=-11$
$\therefore cx ^2+2 ax - b =2 x ^2+12 x +11=2( x +3)^2-7$
$\therefore \text { Minimum value }=-7$