Determine the greatest possible value of

$\sum _{i=1}^{10}\mathrm{cos}3{x}_{i}$

for real numbers$x}_{1},{x}_{2}\dots {x}_{10$ satisfying

$\sum _{i=0}^{10}{\mathrm{cos}x}_{i}=0$

My attempt:

$\sum \mathrm{cos}3x=\sum 4{\mathrm{cos}}^{3}x-\sum 3\mathrm{cos}x=4\sum {\mathrm{cos}}^{3}x$

So now we have to maximize sum of cubes of ten numbers when their sum is zero and each lie in interval [−1,1]. i often use AM GM inequalities but here are 10 numbers and they are not even positive. Need help to how to visualize and approach these kinds of questions.

for real numbers

My attempt:

So now we have to maximize sum of cubes of ten numbers when their sum is zero and each lie in interval [−1,1]. i often use AM GM inequalities but here are 10 numbers and they are not even positive. Need help to how to visualize and approach these kinds of questions.