Question

One root of $x^3-7x+7=0$ lies between

• A. $-1$ and $0$
• B. $-2$ and $-1$
• C. $-3$ and $-2$
• D. $4$ and $-3$

Answer 1

The correct option is D $4$ and $-3$

Fact: If one (odd number) root of $f\left(x\right)$ lie between $\alpha$ and $\beta$ then $f\left(\alpha \right)\cdot f\left(\beta \right)<0$
Here $f\left(x\right)=x^3-7x+7$
$f(-1)=13,f\left(0\right)=7,f(-2)=13,f(-3)=1,f(-4)=-29$
Clearly $f(-3)\cdot f(-4)<0\Rightarrow$ one root of $x^3-7x+7=0$ lie in $-4$ and $-3$