Let e($\lambda$) be the eccentricity of the ellipse $\frac{x^2}{a^2+\lambda }+\frac{y^2}{b^2+\lambda }=1$, where a > b,$\lambda \ge$0. Then

Here,

$e\left(\lambda \right)=\sqrt{1-\frac{b^2+\lambda }{a^2+\lambda }}=\sqrt{\frac{a^2-b^2}{a^2+\lambda }}$

$\Rightarrow e^{\mathrm{\prime }}\left(\lambda \right)=\frac{\sqrt{a^2-b^2}}{{\left(a^2+\lambda \right)}^{\frac{3}{2}}}\left(-\frac{1}{2}\right)<0\mathrm{\forall }\lambda >0$

$\Rightarrow e\left(\lambda \right)$ decreases in the interval $[0,\mathrm{\infty })$

$\underset{\lambda \ge 0}{max} e\left(\lambda \right)=e\left(0\right)=\sqrt{1-{\left(\frac{b}{a}\right)}^2}$

As e($\lambda$) decreases in the interval $[0,\mathrm{\infty })$, e($\lambda$) has no minimum value.