Question

If f(x) is a polynomial satisfying the relation $f\left(x\right)+f\left(2x\right)=5x^2-18$ then $f^1\left(1\right)$ is equal to

• A. 1
• B. 3
• C. Cannot be found since degree of f(x) is not given
• D. 2

Answer 1

The correct option is D 2

Let $f\left(x\right)=a{x}^2+bx+c.....\left(A\right)\left(Byhypothesis\right)$

$\Rightarrow f\left(2x\right)=4a{x}^2+2bx+c\Rightarrow f\left(x\right)+f\left(2x\right)=5a{x}^2+3bx+2c....\left(1\right)$

Given, $f\left(x\right)+f\left(2x\right)=5x^2-\mathrm{18.....}\left(2\right)$

From (1) and (2)

$5a{x}^2+3bx+2c=5x^2-18$

By comparing above equation we get,

$a=1,b=0,c=-9$...........(B)

$\Rightarrow f\left(x\right)=x^2-9$ [From (A) and (B)]

Differentiate above equation with.r.t.$x$

$f^1\left(x\right)=2x$

$\therefore f^1\left(1\right)=2$.

Hence correct answer is option (B)