Q34 / 90 · KVPY 2011 (SB/SX) · Mathematics

Let $f(x)=x^{3}+a x^{2}+b x+c$, where $a, b, c$ are real numbers. If $f(x)$ has a local minimum at $x=1$ and a local maximum at $x=-\frac{1}{3}$ and $f(2)=0$, then $\int_{-1}^{1} f(x) d x$ equals-

  • A
    $\frac{14}{3}$
  • Correct answer
    B
    $\frac{-14}{3}$
  • C
    $\frac{7}{3}$
  • D
    $\frac{-7}{3}$

Solution

$f^{\prime}(x)=3\left(x^{2}-\frac{2}{3} x-\frac{1}{3}\right)=3 x^{2}-2 x-1$ $f(x)=x^{3}-x^{2}-x+\lambda$ $f(2)=8-4-2+\lambda=0 \Rightarrow \lambda=-2$ $f(x)=x^{3}-x^{2}-x-2$ $\int_{-1}^{1} f(x) d x=-2 \int_{0}^{1}\left(x^{2}+2\right) d x=-2\left(\frac{1}{3}+2\right)=\frac{-14}{3}$

Hence, the correct answer is option: 'B'.

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