Q41 / 90 · JEE Main 2017 (09 Apr Online) · Mathematics

Let $f$ be a polynomial function such that $f (3 x) = f^{'} (x) . f^{''} (x),$ for all $x \in R .$ Then :

A
$f (2) + f^{'} (2) = 28$
Correct answer
B
$f^{''} (2) - f^{'} (2) = 0$
C
$f (2) - f^{'} (2) + f^{''} (2) = 10$
D
$f^{''} (2) - f (2) = 4$

Solution

Degree of

f (x)

will be

3

f (x) = a x^{3} + b x^{2} + c x + d

f (3 x) = 27 a x^{3} + 9 b x^{2} + 3 c x + d

f^{'} (x) = 3 a x^{2} + 2 b x + c

f^{''} (x) = 6 a x + 2 b

f (3 x) = f^{'} (x) f^{''} (x)

Comparing the coefficient, we get

27 a = 18 a^{2}

\Rightarrow a = \frac{3}{2}

Also

b = 0, c = 0, d = 0

f (x) = \frac{3}{2} x^{3}

,

f (2) = 12

\Rightarrow f^{'} (x) = \frac{9}{2} x^{2}, f^{''} (x) = 9 x

Hence,

f^{'} (2) = 18, f^{''} (2) = 18

Hence,

f^{''} (2) - f' (2) = 0

Hence, the correct answer is option: 'B'.

Relevant Questions

The temperature T t of a body at time t = 0 is 160 ° F and it decreases continuously as per the differential equation d T d t = − K T − 80 ,31 Jan, 2024 (E)Applications of DE
Let the tangent at any point P on a curve passing through the points 1 , 1 and 1 10 , 100 , intersect positive x -axis and y -axis at the po10 Apr, 2023 (E)Applications of DE
Let a smooth curve y = f x be such that the slope of the tangent at any point x , y on it is directly proportional to - y x . If the curve p25 Jul, 2022 (E)Applications of DE
Let the slope of the tangent to a curve y = f x at x , y be given by 2 tan x cos x - y . if the curve passes through the point 4 , 0 , then 28 Jun, 2022 (E)Applications of DE
Let d y d x = a x - b y + a b x + c y + a , where a , b , c are constants. represent a circle passing through the point 2 , 5 . Then the sho27 Jun, 2022 (M)Applications of DE
The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 unit24 Jun, 2022 (M)Applications of DE