L.H.L. = e² + b = 8k = R.H.L.   $\Rightarrow \mathrm{b}=\frac{-{\mathrm{e}}^2}{3},\mathrm{k}=\frac{{\mathrm{e}}^2}{12}$
L.H.D = e² = 12k = R.H.D.

$\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}{\mathrm{e}}^{\mathrm{x}}-\frac{{\mathrm{e}}^2}{3};\mathrm{x}\le 2\\ \frac{{\mathrm{e}}^2}{12}{\mathrm{x}}^3;\mathrm{x}>2\end{array}$

$\mathrm{g}\text{'}\left(\frac{2}{3}{\mathrm{e}}^2\right)=\frac{1}{\mathrm{f}\text{'}\left(2\right)}=\frac{1}{{\mathrm{e}}^2}$

$\mathrm{g}\text{'}\text{'}\left(\frac{2}{3}{\mathrm{e}}^2\right)=-\frac{\mathrm{f}"\left(2\right)}{{\left(\mathrm{f}\text{'}\left(2\right)\right)}^3}=\frac{-{\mathrm{e}}^2}{{\left({\mathrm{e}}^2\right)}_{}^3}=\frac{-1}{{\mathrm{e}}^4}$