Screenshot_1.png Let f be a polynomial function such that f(3x)=f'(x)f"(x) for x belongs to R Then
@Priti
If f(x) is of first degree its second derivative 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≥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 of degree (n−1)+(n−2) and as two polynomials can be equal for every x only if they have the same degree:
that is f(x) must be of third degree
Equating the coefficients of the same degree we get:
The polynomial which satisfies the equation is then
f (x) = a.r3 given, f(3x) = f'(x) replace, x —+ 3x 270 = = 3/2 similarly, b, c, d become 0
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