Question

If the graph of the function $f\left(x\right)=a{x}^3+x^2+bx+c$ is symmetric about the line $x=2$ , then the value of $a+b$ is equal to

• A. $10$
• B. $-4$
• C. $16$
• D. $-10$

Answer 1

Answer: (B)

$f\left(x\right)$ is symmetric about the line $x=2$
$\therefore f\left(2+x\right)=f\left(2-x\right)$
$a{\left(2+x\right)}^3+{\left(2+x\right)}^2+b\left(2+x\right)+c=a{\left(2-x\right)}^3+{\left(2-x\right)}^2+b\left(2-x\right)+c$
$a\left\{{\left(2+x\right)}^3-{\left(2-x\right)}^3\right\}+\left\{{\left(2+x\right)}^2-{\left(2-x\right)}^2\right\}+b\left\{\left(2+x\right)-\left(2-x\right)\right\}=0$
$a\left\{\left(8+12x+6x^2+x^3\right)-\left(8-12x+6x^2-x^3\right)\right\}+2\left(4x\right)+b\left(2x\right)=0$
$a\left(24x+2x^3\right)+8x+2bx=0$
$2a{x}^3+\left(24a+2b+8\right)x=0$
Which must be true $\forall x\in R$
$\therefore$ it is an identity
$\therefore 2a=0$ , $24a+2b+8=0$
$a=0,2b+8=0$
$b=-4$