Question

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