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1.Relation and Function
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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
A
$[-1,1]$
B
$(-1,1]$
C
$(-1,1)$
D
$\left(-\frac{1}{2}, \frac{1}{2}\right)$
(KVPY-2016)
Solution
(c)
We have,
$P(x)=4 x^3-3 x x \in\left(-\frac{1}{2}, \frac{1}{2}\right)$
$P^{\prime}(x)=12 x^2-3=3\left(4 x^2-1\right)$
$P^{\prime}(x)<0 x \in\left(-\frac{1}{2}, \frac{1}{2}\right)$
Standard 12
Mathematics