$\text{ If }{e}^{\lambda }\text{ and }{e}^{-\lambda }\text{ are the roots of }3x^2-(a+b)x+2a=0,a,b\text{ , }\lambda \in R,\lambda \ne 0\text{ then least integral value of }b\text{ is }$

$f\left(x\right)=3x^2-(a+b)x+2a=0\text{ has roots }{e}^{\lambda }\text{ and }{e}^{-\lambda }\text{ Product of the roots }=1$

$\begin{array}{l}\Rightarrow \frac{2a}{3}=1\Rightarrow a=\frac{3}{2}\\ \text{ Now, sum of the roots }=e^{\lambda }+e^{-\lambda }=\frac{a+b}{3}>2\\ \Rightarrow b>6-a\\ \Rightarrow b>6-\frac{3}{2}\Rightarrow b>\frac{9}{2}\end{array}$

$\text{ Therefore, least integral value of }b\text{ is }5\text{ . }$