Question

If the equation $x^{3}-6x^{2}+9x+\lambda =0$ has exactly one root in $\left(1,3\right),$ then $\lambda $ belongs to the interval

• A. $\left(- 6 , - 3\right)$
• B. $\left(- 4,0\right)$
• C. $\left(- 2,2\right)$
• D. $\left(- 1,3\right)$

Answer 1

Answer: (B)

Let, $f\left(x\right)=x^{3}-6x^{2}+9x+\lambda $
$\Rightarrow f^{'} \left(x\right) = 3 x^{2} - 12 x + 9 = 3 \left(x^{2} - 4 x + 3\right)$
$=3\left(x - 1\right)\left(x - 3\right)$

$f^{'}\left(x\right) < 0,\forall x\in \left(1 , 3\right)f\left(1\right)=4+\lambda ,f\left(3\right)=\lambda $

For $f\left(x\right)=0$ to have exactly one root in $\left(1,3\right),$ the values of $f\left(1\right)$ and $f\left(3\right)$ should have opposite signs
$\Rightarrow f\left(1\right)\cdot f\left(3\right) < 0$
$\Rightarrow \left(\lambda + 4\right)\lambda < 0$
$\Rightarrow \lambda \in \left(- 4,0\right)$