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