Let f ( x ) = x 3 + x 2 f ' ( 1 ) + x f ' ' ( 2 ) + f ' ' ' ( 3 ) , ? x ? R . Then f ' ( 10 ) is equal to _____ . [2024]

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Solution

Q.
$Let f (x) = x^{3} + x^{2} f' (1) + x f'' (2) + f''' (3), x \in R . Then f' (10) is equal to _____ .$ [2024]

Ans.
(202)

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

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

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

$f''' (x) = 6 \Rightarrow f''' (3) = 6$

$f'' (2) = 6 (2) + 2 f' (1) = 12 + 2 f' (1),$

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

$\Rightarrow f' (1) = 3 + 2 f' (1) + 12 + 2 f' (1)$

$\Rightarrow 3 f' (1) = - 15 \Rightarrow f' (1) = - 5$

$∴ f'' (2) = 12 + 2 (- 5) = 2$

Now, $f' (10) = 3 {(10)}^{2} + 2 (10) f' (1) + f'' (2)$

$= 300 + 20 (- 5) + 2 = 202$

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