Multiple choice

Let $f(x)$ is a quadratic function such that $f(0) = 1 ,f(1)=7,f(-1)=1$ and $\displaystyle {\int \frac {f(x) dx}{x^2 (x\, +\, 1)^3}}$ is a rational function. Find the value of $f'(0)$.

  1. $3$
  2. $1$
  3. $2$
  4. $6$
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A Correct answer
Explanation

Given f(0)=1, f(1)=7, f(-1)=1, let f(x) = ax^2 + bx + c. From f(0)=1, c=1. From f(1)=a+b+1=7, a+b=6. From f(-1)=a-b+1=1, a-b=0. Solving gives a=3, b=3. Thus f(x) = 3x^2 + 3x + 1. Then f'(x) = 6x + 3, so f'(0) = 3.

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