MCQ
Let $f$ be a polynomial function such that $f(3x)\, = f'(x) , f''(x)$, for all $x \in R$. Then
- A$f(2) + f'(2)\,= 28$
- ✓$f''(2) -f'(2)\, = 0$
- C$f''(2)-f'(2)\,= 4$
- D$f(2) -f'(2) + f''(2)\, = 10$
$f\left( {3x} \right) = 27a{x^3} + 9b{x^2} + 3cx + d$
$f'\left( x \right) = 3a{x^2} + 2bx + c$
$f''\left( x \right) = 6ax + 2b$
$f\left( {3x} \right) = f'\left( x \right)f''\left( x \right)$
$27a = 18{a^2}$
$a = \frac{3}{2},b = 0,c = 0$
$f\left( x \right) = \frac{3}{2}{x^3}$
$f'\left( x \right) = \frac{9}{2}{x^2},f''\left( x \right) = 9x$
$f'\left( 2 \right) = 18$
$f''\left( 2 \right) = 18$
$ \Rightarrow f''\left( 2 \right) - f\left( 2 \right) = 0$
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If $g: S \rightarrow R$ be defined as $g(x)=\log _{e} f(x),$ then the value of $\mid g "(5)- g "(1) \mid$ is equal to :
$\left[\begin{array}{cc}
2 a+b & a-2 b \\
5 c-d & 4 c+3 d
\end{array}\right]=\left[\begin{array}{cc}
4 & -3 \\
11 & 24
\end{array}\right]$