Question

Let $P ( x )$ be quadratic polynomial with real coefficients such that for all real $x$ the relation $2(1+ P ( x ))= P ( x -1)+ P ( x +1)$ holds. If $P (0)=8$ and $P (2)=32$ then
If the range of $P ( x )$ is $[ m , \infty)$ then the value of ' $m$ ' is

• A. -12
• B. -15
• C. -17
• D. -5

Answer 1

Answer: (C)

Let$ P ( x )= ax ^2+ bx + c $
$P (0)= c \Rightarrow c =8$
also $P (2)=32 \Rightarrow 4 a +2 b +8=32 \Rightarrow 2 a + b =12$
and $P (1)=19 \Rightarrow a + b + c =19 \Rightarrow a + b +8=19 \Rightarrow a + b =11$
$a =1$ and $b =10$
$P(x) =x^2+10 x+8 $
$ =(x+5)^2-17$
$\left.\therefore P ( x )\right|_{\min }=-17 \Rightarrow m =-17$