R 6.77 Let f: R (1) 30 be a functio...
Question

# R 6.77 Let f: R (1) 30 be a function such that f(x)=x3 + x2 f'(1)+ xf"(2)+f''(3), X ER. Then f(2) equals (2) - 2 (3) - 4 (4) 8

JEE/Engineering Exams
Maths
Solution
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( f(x)=x^{3}+x^{2} f^{prime}(1)+x f^{prime prime}(2)+f^{prime prime prime}(3) ) ( Rightarrow f^{prime}(x)=3 x^{2}+2 x f^{prime}(1)+f^{prime prime}(2) quad-C ) ( Rightarrow f^{prime prime}(x)=6 x+2 f^{prime}(1) ) ( Rightarrow f^{prime prime prime}(x)=6 ) ( Rightarrow quad f^{prime prime prime}(3)=6 ) Put ( x=2 ) in ( e q^{n}(2) ) ( f^{prime prime}(2)=12+^{prime} 2 f^{prime}(1) ) Put ( x=1 quad ) in ( quad e q^{n}left(1, Rightarrow f^{prime}(1)=3+2 f^{prime}(1)+f^{prime prime}(2)right. ) ( Rightarrow quad 3+f^{prime}(1)+12+2 f^{prime}(1)=0 ) ( Rightarrow quad 3 f^{prime}(1)=-15 ) ( Rightarrow quad f^{prime}(1)=-5 ) ( Rightarrow f^{prime prime}(2)=12+2(-5)=12-10=2 ) ( Rightarrow f(x)=x^{3}-5 x^{2}+2 x+6 ) Now, ( f(2)=8-5(4)+4+6=8-20+10=-2 ) Ophon (2) is corret