Let $f(x) = x^{3} + x^{2}f'(1) + 2xf^{''}(2) + f^{'''}(3)$, $x \in R$. Then

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Let $f(x) = x^{3} + x^{2}f'(1) + 2xf^{''}(2) + f^{'''}(3)$, $x \in R$. Then the value of $f'(5)$ is :

Accepted Answer

$f'(x) = 3x^{2} + 2{xf}'(1) + 2f^{''}(2)$

$${f^{''}(x) = 6x + 2f'(1) }{f^{''}(2) = 12 + 2f'(1) }{\therefore f'(x) = 3x^{2} + 2{xf}'(1) + 2\left( 12 + 2f'(1) \right) }{f'(x) = 3x^{2} + 2(x + 2)f'(1) + 24 }$$Putting, $x = 1$

$${f'(1) = 3 + 6f'(1) + 24 }{- 5f'(1) = 27 \Rightarrow f'(1) = \frac{- 27}{5} }{\therefore f^{''}(2) = 12 + 2\left( \frac{- 27}{5} \right) = 12 - \frac{54}{5} = \frac{6}{5} }{\therefore f'(x) = 3x^{2} - \frac{54}{5}x + \frac{12}{5} }{\therefore f'(5) = 75 - 54 + \frac{12}{5} = \frac{117}{5}}$$

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