Let f,:,R to R be a function such t | vedclass

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Question

$f\left( x \right){x^3} + a{x^2} + bx + c$

$f'\left( x \right) = 3{x^2} + 2ax + b \Rightarrow c = 6$

$f''\left( x \right) = 6x + 2a$

Vedclass · Accepted Answer

$-2$

Vedclass · Answer

$f\left( x \right){x^3} + a{x^2} + bx + c$

$f'\left( x \right) = 3{x^2} + 2ax + b \Rightarrow c = 6$

$f''\left( x \right) = 6x + 2a$

$f'''\left( x \right) = 6\,\,a = f'\left( 1 \right) = 3 + 2a + b\,\,\,\,\, \Rightarrow a + b =  - 3$

$b = f'' = 12 + 2a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow 2a - b =  - 12$

$c = f'''\left( 3 \right)$           $ \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$

Let $f\,:\,R \to R$ be a function such that $f\left( x \right) = {x^3} + {x^2}f'\left( 1 \right) + xf''\left( 2 \right) + f'''\left( 3 \right)$, $x \in R$. Then $f(2)$ equals

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