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

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

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
$\Rightarrow f\left(1\right)\cdot f\left(3\right)<0$
$\Rightarrow \left(\lambda +4\right)\lambda <0$
$\Rightarrow \lambda \in \left(-4,0\right)$