Question

Consider a quadratic polynomial f(x) satisfying $f\left(x\right)\ge 1\forall x\in R$ , $f\left(2\right)$ = $1$ and $f\left(3\right)=3$. Then the value of $f\left(5\right)$ is equal to-

• A. $20$
• B. $10$
• C. $19$
• D. $13$

Answer 1

The correct option is C $19$

$f\left(2\right)=1f\left(3\right)=3$

$f\left(x\right)=a{x}^2+bx+c\to$ quadratic polynomial

$f\left(x\right)\ge 1\forall x\in R$

$\Rightarrow f\left(x\right)=p(x-q{)}^2+1[\because p(x-q{)}^2\ge 0forp>0]$

$f\left(2\right)=p(2-q{)}^2+1=1\Rightarrow p(2-q{)}^2=0\Rightarrow$ or $=2(\because p>0)$

$f\left(3\right)=p(3-q{)}^2+1=3\Rightarrow p+1=3\Rightarrow p=2$

$f\left(x\right)=2(x-2{)}^2+1$

$f\left(5\right)={2.3}^2+1=19$