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Question

Let f be a polynomial function such that f(3x)=f(x)f′′(x), for all xR. Then:

A
f(2)+f(2)=28
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B
f′′(2)f(2)=0
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C
f′′(2)f(2)=4
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D
f(2)f(2)+f′′(2)=10
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Solution

The correct option is B f′′(2)f(2)=0
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 n2. Then f(x) will have degree (n1) and f′′(x) degree (n2).
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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