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:


ANSWER

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


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
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