Let f ( x ) = x 3 + x 2 f ' ( 1 ) + 2 x f ' ' ( 2 ) + f ' ' ' ( 3 ) , x. Then the value of f'(5) is [2026]

Question

Let $f (x) = x^{3} + x^{2} f' (1) + 2 x f'' (2) + f''' (3), x \in R .$ Then the value of $f' (5)$ is [2026]

Smart Achievers · Accepted Answer

(1)

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

$f'' (x) = 6 x + 2 f' (1)$

$f'' (2) = 12 + 2 f' (1)$

$∴$ $f' (x) = 3 x^{2} + 2 x f' (1) + 2 (12 + 2 f' (1))$

$f' (x) = 3 x^{2} + 2 (x + 2) f' (1) + 24$

Putting $x = 1$

$f' (1) = 3 + 6 f' (1) + 24$

$- 5 f' (1) = 27 \Rightarrow f' (1) = \frac{- 27}{5}$

$∴$ $f'' (2) = 12 + 2 (\frac{- 27}{5}) = 12 - \frac{54}{5} = \frac{6}{5}$

$∴$ $f' (x) = 3 x^{2} - \frac{54}{5} x + \frac{12}{5}$

$∴$ $f' (5) = 75 - 54 + \frac{12}{5} = \frac{117}{5}$

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