Question

Let $f$ be a polynomial function such that $f (3x)= f' (x) ⋅ f" (x)$, for all $x \epsilon R$. Then :

• A. $f (2) + f' (2) = 28$
• B. $f " (2) - f ' (2) = 0$
• C. $f" (2) - f (2) = 4$
• D. $f (2) - f ' (2) + f " (2) = 10$

Answer 1

Answer: (B)

$f(x) = ax^3 + bx^2+ cx + l $
$f(3x) = 27ax^3+ 9bx^2 + 3cx + d $
$f'(x) = 3ax^2 + 2bx + c $
$f"(x) = 6ax + 2b$
$f(3x) = f'(x) f"(x) $
$27a = 18a^2$
$a=\frac{3}{2}, $
$b=0, $
$c=0, $
$d=0,$
$f \left(x\right)=\frac{3}{2}x^{3}$
$f \left(x\right)=\frac{9}{2}x^{2}, f ''\left(x\right)=9x$