Using Kuhn-Tucker conditions, find the value(s) of \( \beta \) for which the point \( x_{1}^{*}=1, x_{2}^{*}=2 \) will be optimal to the problem:
\[
\text { Maximize } f\left(x_{1}, x_{2}\right)=2 x_{1}+\beta x_{2}
\]
subject to
\[
\begin{array}{l}
g_{1}\left(x_{1}, x_{2}\right)=x_{1}^{2}+x_{2}^{2}-5 \leq 0 \\
g_{2}\left(x_{1}, x_{2}\right)=x_{1}-x_{2}-2 \leq 0
\end{array}
\]
Verify your result using a graphical procedure.
