What is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$?

No need for calculus here; by AM-GM we have $1+a_1\ge2\sqrt{a_1},\ldots, 1+a_{n}\ge 2\sqrt{a_n}$ and multiplying them yields $$\prod_{1\le i\le n} (1+a_i)\ge 2^{n}\sqrt{a_1a_2\cdots a_n}=2^n,$$ with equality when $a_1=\cdots = a_n = 1$. In general if $a_1a_2\cdots a_k = M$ for some positive real $M$, then we can give a lower bound on the minimum value, which is similarly computed to be $2^n\cdot \sqrt{M}$, with equality at $a_1=\cdots = a_n = \sqrt[n]{M}$. The absolute minimum, as discussed below, is $(1+\sqrt[n]{M})^n$.

Use the fact that $$1+a_k \geq 2\sqrt{a_k}$$ to conclude what you want.