If f ( x ) = x 2 + g ' ( 1 ) x + g ' ' ( 2 ) and g ( x ) = f ( 1 ) x 2 + x f ' ( x ) + f ' ' ( x ) , then the value of f ( 4 ) - g ( 4 ) is equal to [2023]

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Solution

Q.
If $f (x) = x^{2} + g' (1) x + g'' (2)$ and $g (x) = f (1) x^{2} + x f' (x) + f'' (x)$ , then the value of $f (4) - g (4)$ is equal to [2023]

Ans.
(14)

$Let g' (1) = A$

$g'' (2) = B, f (x) = x^{2} + A x + B, f (1) = A + B + 1$

$f' (x) = 2 x + A, f'' (x) = 2$ ,

$g (x) = (A + B + 1) x^{2} + x (2 x + A) + 2$

$\Rightarrow g (x) = x^{2} (A + B + 3) + A x + 2$

$g' (x) = 2 x (A + B + 3) + A, g' (1) = A$

$\Rightarrow 2 (A + B + 3) + A = A$

$A + B = - 3 ...(i)$

$g'' (x) = 2 (A + B + 3), g'' (2) = B$

$\Rightarrow 2 (A + B + 3) = B$

$\Rightarrow 2 A + B = - 6 ...(ii)$

$From (i) and (ii): A = - 3, B = 0$

$f (x) = x^{2} - 3 x, f (4) = 16 - 12 = 4$

$g (x) - 3 x + 2, g (4) = - 12 + 2 = - 10$

$f (4) - g (4) = 4 - (- 10) = 14$

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