If f(x)=x3−x2f'(1)+xf''(2)−f'''(3),x∈R , then

Let f(x)=x3+bx2+cx+df1(x)=3x2+2bx+cf11(x)=6x+2bf111(x)=6Given, f(x)=x3−x2f1(x)+xf11(2)−f111(3)x3+bx2+cx+d=x3−x2(3+2b+c)+x(12+2b)−6By comparing coefficients−b=3+2b+c c=12+2b−3=3b+c……(1) 2b−c=−12 Eq. (1) \& (2) 3b+c=−32b−c=−12−⋯5b=−15b=−3,c=6,d=−6f(x)=x3−3x2+6x−6From option 22f(0)−f(1)+f(3)=f(2)2(−6)−(−2)+12=2

If f(x)=x3−x2f'(1)+xf''(2)−f'''(3),x∈R , then

Let f(x)=x3+bx2+cx+df1(x)=3x2+2bx+cf11(x)=6x+2bf111(x)=6Given, f(x)=x3−x2f1(x)+xf11(2)−f111(3)x3+bx2+cx+d=x3−x2(3+2b+c)+x(12+2b)−6By comparing coefficients−b=3+2b+c c=12+2b−3=3b+c……(1) 2b−c=−12 Eq. (1) \& (2) 3b+c=−32b−c=−12−⋯5b=−15b=−3,c=6,d=−6f(x)=x3−3x2+6x−6From option 22f(0)−f(1)+f(3)=f(2)2(−6)−(−2)+12=2

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

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$3 f (1) + f (2) = f (3)$

$2 f (0) - f (1) + f (3) = f (2)$

$f (3) - f (2) = f (1)$

$f (1) + f (2) + f (3) = f (0)$

answer is B.

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Detailed Solution

$Let f (x) = x^{3} + {bx}^{2} + cx + d$
$\begin{array}{l} f^{1} (x) = 3 x^{2} + 2 bx + c \\ f^{11} (x) = 6 x + 2 b \\ f^{111} (x) = 6 \end{array}$
Given, $f (x) = x^{3} - x^{2} f^{1} (x) + {xf}^{11} (2) - f^{111} (3)$
$x^{3} + {bx}^{2} + cx + d = x^{3} - x^{2} (3 + 2 b + c) + x (12 + 2 b) - 6$
By comparing coefficients
$\begin{array}{l} - b = 3 + 2 b + c c = 12 + 2 b \\ - 3 = 3 b + c \dots \dots (1) 2 b - c = - 12 \\ Eq. (1) \& (2) \\ 3 b + c = - 3 \\ 2 b - c = - 12 \\ - \dots \\ 5 b = - 15 \end{array}$
$\begin{aligned} b = - 3, c = 6, d = - 6 \\ f (x) = x^{3} - 3 x^{2} + 6 x - 6 \end{aligned}$
From option 2
$\begin{aligned} 2 f (0) - f (1) + f (3) = f (2) \\ 2 (- 6) - (- 2) + 12 = 2 \end{aligned}$