Let $x_1,x_2,...,x_n$ be positive real numbers such that their product is $1$. The minimum value of $(1+x_1)(1+x_2)...(1+x_n)$ is:

Solution

1. By the AM-GM inequality, $1+x_i\ge 2\sqrt{x_i}$ for each $i$.
2. Multiplying these inequalities for $i=1$ to $n$, we get ${\prod }_{i=1}^n(1+x_i)\ge 2^n\sqrt{{\prod }_{i=1}^n{x}_i}$.
3. Since the product of $x_i$ is 1, the expression is at least $2^n$.
4. Equality holds when $$\begin{array}{rl}{x}_1& =x_2\\ & =...\\ & =x_n\\ & =1\end{array}$$.
5. Hence the answer is (A).