A) 30
B) - 2
C) - 4
D) 8
Correct Answer: B
Solution :
\[f(x)={{x}^{3}}+{{x}^{2}}f'(1)+xf\text{''}\,(2)+f'\text{''}(3),x\in R\] \[f\left( x \right)=3{{x}^{2}}+2xf\left( 1 \right)+f''\left( 2 \right)\] \[\Rightarrow \,\,\,f\left( 1 \right)=3+2f\left( 1 \right)+f''\left( 2 \right)\] \[\Rightarrow \,\,\,\,f\left( 1 \right)+f''\left( 2 \right)+3=0\] \[f''\left( x \right)=6x+2f\left( 1 \right)\Rightarrow f''\left( 2 \right)=12+2f\left( 1 \right)\] \[\Rightarrow \,\,-f\left( 1 \right)-3=\text{ }12+2f\left( 1 \right)\] \[=f'(1)=-5;\,\,\,f\text{''}(2)=2\] \[f'''\left( x \right)=6\,\,\Rightarrow \,\,f'''\left( 3 \right)=6\] \[f\left( x \right)={{x}^{3}}-5{{x}^{2}}+2x+6\] \[f\left( 2 \right)=8-20+4+6=-2\]
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