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1.Relation and Function
hard
माना फलन $f : R \rightarrow R$ इस प्रकार है कि $f ( x )= x ^{3}+ x ^{2} f ^{\prime}(1)+ xf ^{\prime \prime}(2)+ f ^{\prime \prime \prime}(3), x \in R$ तो $f(2)$ बराबर है
A
$-4$
B
$30$
C
$-2$
D
$8$
(JEE MAIN-2019)
Solution
$f\left( x \right){x^3} + a{x^2} + bx + c$
$f'\left( x \right) = 3{x^2} + 2ax + b \Rightarrow c = 6$
$f''\left( x \right) = 6x + 2a$
$f'''\left( x \right) = 6\,\,a = f'\left( 1 \right) = 3 + 2a + b\,\,\,\,\, \Rightarrow a + b = – 3$
$b = f'' = 12 + 2a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow 2a – b = – 12$
$c = f'''\left( 3 \right)$ $ \Rightarrow c = 6$ and $a = – 5,b = 2$
$ \Rightarrow f\left( x \right) = {x^2} – 5{x^2} + 2x + 6$
$ \Rightarrow f\left( 2 \right) = 8 – 20 + 4 + 6 = – 2$
Standard 12
Mathematics
