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EEJ MAIN Mathematics QUESTION #6032
Question 1

Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$ for all $x\in\mathbb{R}$. Find $f(2)$.

  • -4✔️
  • 30
  • -2
  • 8
Correct Answer Explanation

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

$f'(x)=3x^2+2ax+b \Rightarrow f'(1)=3+2a+b=a \Rightarrow a+b=-3$ ...(i)

$f''(x)=6x+2a \Rightarrow f''(2)=12+2a=b$ ...(ii)

$f'''(x)=6 \Rightarrow f'''(3)=6=c$ ...(iii)

From (ii): $b=12+2a$. Sub in (i): $a+12+2a=-3\Rightarrow3a=-15\Rightarrow a=-5$, $b=2$, $c=6$.

$f(2)=8+4(-5)+2(2)+6=8-20+4+6=\mathbf{-2}$

Wait: $f(2)=8-20+4+6=-2$. So correct index is 2 (option C: $-2$).