Mathematics

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

$f''\left( 2 \right) - f'\left( 2 \right) = 0$

##### SOLUTION
The correct answer is $B$
Now,
If $f(x)$ is of first degree its second derivatives is identically null, so also $f(x)$ would have to be identically null.
To satisfy the equation $f(3x)=f'(x)f''(x)$
Let, then $f(x)$  be a generic polynomial  of degree $n \ge 2$. Then $f'(x)$ will have degree $(n-1)$ and $f''(x)$ degree $(n-2)$.
Now, The product $f'(x).f"(x)$ is a polynomial can be equal  for every $x$ only is they the same degree
$f(3x)=f'(x)f''(x)$.

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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