Let $n \in \mathbb{N}$, $n \geq 2$. Let $x_1, \ldots, x_n \in (0, \pi)$. Set $x = \frac{(x_1 + \cdots + x_n)}{n}$. Which of the following statements are true?
(b) $\prod_{k=1}^n \sin x_k \leq \sin^n x$
Option b is correct. We proceed by induction. Take $n = 2$. Then, we have $$\sin x_1+\sin x_2 = \frac{1}{2} \cos(x_1-x_2)- \cos(x_1+x_2)) \leq \frac{1-\cos(x_1+x_2)}{2} = \sin^2 x$$ as $\left|\cos x\right| \leq 1$. How to prove further?
Yes, it's true.
Let $f(x)=\ln\sin{x}$
Thus, $$f''(x)=-\frac{1}{\sin^2{x}}<0,$$ which says that $f$ is a concave function and your inequality it's just Jensen for this function: $$\sum_{k=1}^n\ln\sin{x_k}\leq n\ln\sin{x}$$ or $$\prod_{k=1}^n\sin{x_k}\leq\sin^nx.$$