Question

Suppose $f\left(x\right)=3x^3-13x^2+14x-2$, it is assumed that $f\left(x\right)=0$ will have 3 root say $\alpha ,\beta$ and $\gamma$, where $\alpha <\beta <\gamma$

$\left[\alpha \right],\left[\beta \right],\left[\gamma \right]$ (where, [-] denotes the greatest function) will be in

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is A AP

$f\left(x\right)=3x^3-13x^2+14x-2$

$f\left(0\right)=-2=-ve$

$f\left(1\right)=3-13+14-2=2=+ve$

$f\left(2\right)=24-52+28-2=-2=-ve$

$f\left(3\right)=81-117+42-2=4=+ve$

$\therefore$ One root lies between 0 & 1

One root lies between 1 & 2

One root lies between 2 & 3

$\therefore \alpha \in (0,1)\Rightarrow \left[\alpha \right]=0$

$\beta \in (1,2)\Rightarrow \left[\beta \right]=1$

$\gamma \in (2,3)\Rightarrow \left[\gamma \right]=2$

$\left[\alpha \right],\left[\beta \right],\left[\gamma \right]$ are in AP