Is my claim true about the maximum and minimum of the given function? I have this three-variable function
$$f(x,y,z)=x^9 \cos (2 x y+x z)+\left(5 x^4+1\right) \cos (x y+x z)+\left(5 x^4+3\right)\\ \qquad+\left(4 x^3+1\right) \cos (x y+2 x z)+x^3 \cos (x z)+\left(2 x^2+7\right) \cos (2 x y+2 x z)\\ \qquad+\left(x^2+5\right) \cos (x y)+\left(2 x^2+7\right) \cos (2 x z)\\+\left(3 x^7+5 x^2+1\right) \cos (x y-x z)+(x+1) \cos (2 x y)$$
for $x,y,z>0$.
Now, since all the coefficients of the functions $\cos$ are positive, can I claim that the inequality $f_{-1}\leq f(x,y,z)\leq f_{+1}$ holds, where $f_{\pm1}$ are the function $f(x,y,z)$ when all the $\cos$ are $+1$ and $-1$?

Yes, considering that the variables only take positive values, setting all the co-sines to 1 will provide an upper bound and setting all the co-sines equal to -1 will provide a lower bound.
