Consider the entropy defined as

$S(p_1,p_2,$dots,$p_n)=-{\displaystyle \underset{i=1}{\overset{n}{}}}{p}_{i}ln{p}_i$

and the function

$g(p_1,p_2,$dots,$p_n)={\displaystyle \underset{i=1}{\overset{n}{}}}{p}_i$

$(1)(20$ pts$)$ Use the method of Lagrange multipliers to find out the distribution vec$(p)=(p_1,p_2,$dots,$p_n)$ for which $S$ is an extrema under the constrain $g=1.$