Finding maximum value of a trigonometric functions

Question

Let the function $f(x_1, x_2,x_3) = \cos(x_1-x_2) +\cos(x_2-x_3)+\cos(x_3-x_1)$, where $x_1, x_2, x_3$ are distinct real numbers.
Here, the value of minimum two cosine functions is going to be changed if I change the value of either one variable and which would then change the value of my whole function.
So, I can't say that it's maximum value would be $3$ since cosine function has a maximum value of $1$.
Then, how can we find the maximum values of these types of trigonometric functions which are dependent like these?

Answer 1

So, taking all your constraints, I will rewrite the question in my way as,

f(x, y, z) = cos(x - y)+cos(y - z)+cos(z - x)

Where, 0< x, y, z <2π

Let, x - y = A, y - z= B, & z - x = -(A + B)

f(A, B) = cos(A) + cos(B) + cos(A + B)

As cos(-x) = cosx

Keeping in mind that A can be equal to B

Where -2π< A, B <2π, For maxima, A=B=0, but according to your constraint A,B≠0, so we can't take equility.

For minima, all 3 terms should be negative. And sum of their magnitude should be maximum. So,

|cos(A)| = |cos(B)| = |cos(A + B)|

A consequence of AM-GM inequality

Taking magnitudes,

|cos(k)| = |cos(2k)|

So, as k can be 2π/3

As maxima should be on every value of given family

f(A,B) = 3×cos(2π/3) = -3/2,

We are done!

-3/2 ≤ f(A,B) < 3