Let \(f\) be a non-zero polynomial function such that \(f(3x)=f'(x)f''(x)\). Then the value of \(f(6)\) is :
Show Hint
- 648
- 324
- 136
- 108
The Correct Option is B
Solution and Explanation
Let \(f(x)=ax^3+bx^2+cx+d\), \(a\neq0\). Then \(f'(x)=3ax^2+2bx+c\) and \(f''(x)=6ax+2b\).
\(f'(x)f''(x)=18a^2x^3+18abx^2+(4b^2+6ac)x+2bc\).
\(f(3x)=27ax^3+9bx^2+3cx+d\).
Compare: \(27a=18a^2\Rightarrow a=\frac32\). \(9b=18ab=27b\Rightarrow b=0\). \(3c=4b^2+6ac=9c\Rightarrow c=0\). \(d=2bc=0\).
So \(f(x)=\frac32x^3\) and \(f(6)=\frac32\times216=324\).
Answer: (2) 324
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