I want to find the range (maximum and minimum) of this three-variable function
$$f=f(x_1,x_2,x_3)=2+\cos \left(x_1-x_2\right)+\sin \left(x_1+x_2-x_3 \right)+\cos \left(x_1+x_3 \right)$$
where all the three variables $-2\pi<x_i<2\pi$.
Is it sufficient to solve of the system of equation
$$\frac{\partial f}{\partial x_1}=\frac{\partial f}{\partial x_2}=\frac{\partial f}{\partial x_3}=0,$$
or should I consider any other condition as well?
Best Answer
Following Vasya's comment, we observe that the three expressions
$$ x_1-x_2 \\ x_1+x_2-x_3 \\ x_1+x_3 $$
are linearly independent, so we can solve (for example)
\begin{align} x_1-x_2 \phantom{{}+x_3} & = 0 \\ x_1+x_2-x_3 & = \frac\pi2 \\ x_1\phantom{{}+x_2}+x_3 & = 0 \end{align}
to yield $x_1 = x_2 = \pi/6, x_3 = -\pi/6$ to get the maximum of $5$. A similar approach yields the minimum.