Question

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

• A. $[-1,1]$
• B. $(-1,1]$
• C. $(-1,1)$
• D. $(-\frac{1}{2},\frac{1}{2})$

Answer 1

The correct option is C $(-1,1)$

$p\left(x\right)=4x^3-3x{p}^{\prime }\left(x\right)=12x^2-3=12(x^2-\frac{1}{4})$
$p^{\prime }\left(x\right)<0\forall x\in (-\frac{1}{2},\frac{1}{2})$
So $p\left(x\right)$ is a monotonic decreasing function for $x\in (-\frac{1}{2},\frac{1}{2})$

Maximum value $=p\left(\frac{-1}{2}\right)=1$
Minimum value $=p\left(\frac{1}{2}\right)=-1$
Hence the range of the function $\in (-1,1)$