Let f:RR be a function such that f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3), xR.

Question

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

Accepted Answer

Let $f(x)=x^3+ax^2+bx+c$. Then $f'(1)=3+2a+b=a\Rightarrow a+b=-3$ ... (1)

$f''(2)=12+2a=b\Rightarrow2a-b=-12$ ... (2)

$f'''(3)=6=c$

Adding (1) and (2): $3a=-15\Rightarrow a=-5,\ b=2$. So $f(x)=x^3-5x^2+2x+6$, and $f(2)=8-20+4+6=-2$.

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