Q56 / 90 · KVPY 2016 (SB/SX) · Mathematics

The range of the polynomial $p(x)=4 x^{3}-3 x$ as $x$ varies over the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is

  • A
    $[-1,1]$
  • B
    $(-1,1]$
  • Correct answer
    C
    $(-1,1)$
  • D
    $\left(-\frac{1}{\underline{2}}, \underline{\underline{1}}{\underline{2}}\right)$

Solution

$\mathrm{P}^{\prime}(\mathrm{x})=12 \mathrm{x}^{2}-3=3\left(4 \mathrm{x}^{2}-1\right)$ $\mathrm{In}\left(-\frac{1}{2}, \frac{1}{2}\right) \mathrm{P}^{\prime}(\mathrm{x}) < 0$ $\Rightarrow \mathrm{P}(\mathrm{x})$ is decreasing $\Rightarrow$ Range $\in(\mathrm{P}(-1), \mathrm{P}(1))$ $\quad$ Range $\in(-1,1)$

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

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