Prove that the product of sines of angles of a triangle is min if it is equilateral I was solving problems about extreme values and conditional extrema , I encounter this problem : 
Prove that the product of the sines of the three angles in a triangle is 
minimum if the triangle is equilateral .
I started by writing the objective function and differentiating it to find the critical points but I do not know how to complete the solution. 
$$F(x,y,z)=sin(x)sin(y)sin(z)$$
then 
$$F_x= cos(x)sin(y)sin(z)=0$$
hence
$$x=(2n+1)\pi/2 \ \ ,\ \  y=n\pi \ \ ,\ \  z=n\pi$$
where $$n=0,\pm1,\pm2,...$$
Similarly $$F_y=0$$
gives
$$x=n\pi\ \ ,\ \  y=(2n+1)\pi/2  \ \ ,\ \  z=n\pi$$
Finally
$$F_z=0$$
gives
$$x=n\pi \ \ ,\ \  y=n\pi \ \ ,\ \  z=(2n+1)\pi/2 $$
 A: First of all I think the exercise would be to maximise the product, because otherwise the statement is not even true (just take $x$ almost $\pi$ and $y=z=\frac{\pi-x}{2}.$)
In that case there's a more direct approach.
Observe that $\sin$ is concave on $[0,\pi].$ So by applying Jensen we get
\begin{equation}\frac{\sin(x)+\sin(y)+\sin(z)}{3}\leq \sin\left(\frac{x+y+z}{3}\right)=\frac{\sqrt{3}}{2}.
\end{equation}
Using this and AM-GM we conclude
\begin{equation}\sin(x)\sin(y)\sin(z)\leq \left( \frac{\sin(x)+\sin(y)+\sin(z)}{3}\right)^3\leq \frac{9\sqrt{3}}{8}
\end{equation}with equality if and only if $x=y=z$ (due to AM-GM and Jensen).
A: An alternate approach uses the Law of Sines. Minimizing $\sin a \sin b\sin c$ is the same as minimizing $ABC$, since $\frac{A}{\sin a} = \frac{B}{\sin b} = \frac{C}{\sin c}$. Fixing the perimeter at $n$ (since the ratio of the sides is clearly independent of the perimeter for given angles), this is the same as minimizing
$$xyz,\ x+y+z=n,\ x \ge 0, y\ge 0, z\ge 0.$$
It's not hard to see that the maximum occurs for $x=y=z$.
A: We wish to minimise $\ln\sin x +\cdots$ subject to $x+y+z=\pi$ with nonnegative angles. The function $\ln\sin p$ has first derivative $\cot p$ and second derivative $-\csc^2 p<0$, and the desired result follows from Jensen's theorem for concave functions.
