# Let f be a polynomial function such that f (3x)=f '(x) ⋅ f ''(x), for all x belongs to R. Then the correct option is?

Feb 26, 2018

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

and the correct option is $\left(2\right)$

#### Explanation:

If $f \left(x\right)$ is of first degree its second derivative is identically null, so also $f \left(x\right)$ would have to be identically null. to satisfy the equation $f \left(3 x\right) = f ' \left(x\right) f ' ' \left(x\right)$

Let then $f \left(x\right)$ be a generic polynomial of degree $n \ge 2$. Then $f ' \left(x\right)$ will have degree $\left(n - 1\right)$ and $f ' ' \left(x\right)$ degree $\left(n - 2\right)$

Now, the product $f ' \left(x\right) \cdot f ' ' \left(x\right)$ is a polynomial of degree $\left(n - 1\right) + \left(n - 2\right)$ and as two polynomials can be equal for every $x$ only if they have the same degree:

$\left(2\right) \text{ } f \left(3 x\right) = f ' \left(x\right) \cdot f ' ' \left(x\right)$

implies:

$n = \left(n - 1\right) + \left(n - 2\right)$

$n = 2 n - 3$

$n = 3$

that is $f \left(x\right)$ must be of third degree:

$f \left(x\right) = a {x}^{3} + b {x}^{2} + c x + d$

$f ' \left(x\right) = 3 a {x}^{2} + 2 b x + c$

$f ' ' \left(x\right) = 6 a x + 2 b$

then $\left(2\right)$ becomes:

$27 a {x}^{3} + 9 b {x}^{2} + 3 c x + d = \left(3 a {x}^{2} + 2 b x + c\right) \left(6 a x + 2 b\right)$

$27 a {x}^{3} + 9 b {x}^{2} + 3 c x + d = 18 {a}^{2} {x}^{3} + 12 a b {x}^{2} + 6 a c x + 6 a b {x}^{2} + 4 {b}^{2} x + 2 b c$

$27 a {x}^{3} + 9 b {x}^{2} + 3 c x + d = 18 {a}^{2} {x}^{3} + 18 a b {x}^{2} + \left(6 a c + 4 {b}^{2}\right) x + 2 b c$

Equating the coefficients of the same degree we get:

$27 a = 18 {a}^{2}$

and so as $a \ne 0$

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

Then:

$9 b = 18 a b = 27 b$

$b = 0$

at the first degree:

$3 c = 6 a c + 4 {b}^{2}$

$3 c = 9 c$

$c = 0$

and finally:

$d = 2 b c = 0$

The polynomial which satisfies the equation is then:

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

so that:

$f ' \left(x\right) = \frac{9}{2} {x}^{2}$

$f ' ' \left(x\right) = 9 x$

$f \left(2\right) = 12$

$f ' \left(2\right) = 18$

$f ' ' \left(2\right) = 18$

and only the second statement is correct.