I am trying to find and classify the extrema of the following function: $f(x,y,z)=\sin(x)+\sin(y)+\sin(z)-\sin(x+y+z)$, with $0\leq x \leq \pi, 0\leq y \leq \pi, 0\leq z \leq \pi$.
I have found three critical points: $(0,0,0),(\pi,\pi,\pi), (\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$. The hessian matrix only yields a conclusion for the point $(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$, which is a maximum. But for the other points, the hessian matrix is null.
What I usually do in these cases is studying directly the increment of the funcion $f$ in such points. However, I do not manage to manipulate the expression of the increment in such a way that it is clear whether the point is a maximum, a minimum or a saddle point.
If anyone could give me a hint, it would be really helpful.
Thanks.
$$\sum_{cyc}\sin{x}-\sin(x+y+z)\leq1+1+1+1=4.$$ The equality occurs for $x=y=z=\frac{\pi}{2},$ which says that we got a maximal value.
Also, $$\sin{x}+\sin{y}+\sin{z}-\sin(x+y+z)=$$ $$=\sin{x}+\sin{y}+\sin{z}-\sin{x}\cos(y+z)-\cos{x}\sin(y+z)=$$ $$=\sin{x}(1-\cos(y+z))+\sin{y}(1-\cos{x}\cos{z})+\sin{z}(1-\cos{x}\cos{y})\geq0.$$ The equality occurs for $x=y=z=0$ or for $x=y=z=\pi,$ which says that we got a minimal value.
We see that these critical points they are points of the global maximum or of the global minimum.
For finding of the minimal value also Karamata helps:
Let $x\geq y\geq z$.
Thus, since $\sin$ is a concave function on $[0,\pi]$ and $(x+y+z,0,0)\succ(x,y,z),$ we obtain: $$\sin{x}+\sin{y}+\sin{z}\geq\sin(x+y+z)+\sin0+\sin0,$$ which gives $$\sin{x}+\sin{y}+\sin{z}-\sin(x+y+z)\geq0.$$