Let $ n > 1 \in \mathbb{Z^+}$. Consider the following function: $$S(p_1, p_2,...,p_n)=-\sum_{i=1}^n p_i \, ln(p_i) $$
defined for $0 < p_1, p_2,...,p_n < 1$ and subject to the constraint, $$\sum_{i=1}^np_i=1$$
Explain with details whether the function has absolute maximum and minimum. Find the point(s) using the method of Lagrange Multipliers.
How can I solve this question?
Here's how to get started. $$-\nabla \sum_{i=1}^np_i\ln p_i=\lambda \nabla (\sum_{i=1}^np_i-1). $$ $$-[\ln p_1+1,...,\ln p_n+1]=\lambda[1,...,1]$$ $$=[\lambda,...,\lambda].$$ Thus $$-(\ln p_1+1)=\lambda,...,-(\ln p_n+1)=\lambda$$ $$\ln p_1+1=...=\ln p_n+1$$ $$\ln p_1=...=\ln p_n$$ $$p_1=...=p_n$$ $$p_1+...+p_1=1$$ $$p_1=1/n.$$ $$p_1=...=p_n=1/n.$$ The critical point is $(\frac{1}{n},...,\frac{1}{n}).$ You can take it from there.