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