Fin the minimum value of $\displaystyle a$ and $\displaystyle b$ for which $\displaystyle f(x) = x^x$ ; $\displaystyle f : [a,\infty] \rightarrow [b,\infty)$ to be an invertible function.
$\displaystyle f'(x)=x^x(1+\ln x)$
$\displaystyle f'(x)=0$
$\displaystyle x=\frac{1}{e}$
$\displaystyle f\left(\frac{1}{e}\right)=\frac{1}{e^{\frac{1}{e}} }$
Therefore, for f to be invertible,$\displaystyle f:\left[\frac{1}{e},\infty\right)\rightarrow\left[\frac{1}{e^{\frac{1}{e}}},\infty\right)$
Thus $\displaystyle a=\frac{1}{e}$ and $\displaystyle b=\frac{1}{e^{\frac{1}{e}}}$