The difference between the greatest and least values of the function f(x)  =   cos x  +  12   cos 2x   −13    cos  3x  is

The given function is periodic, with period  2π.   So the  difference between the greatest and least values of the function is the difference between these values on the  interval  [0,  2π] . We have f′(x)=−(sin⁡x+sin⁡2x−sin⁡3x)=−4sin⁡xsin⁡(3x/2)sin⁡(x/2) Hence x=0,2π/3,π and 2π are the critical points. Also, f(0)  =  1+1/2−1/3  =  7/6,  f(2π/3)  =−13/12,  f(π)  =  −1/6  and  f(2π)  =  7/6.Hence the greatest value is 7/6 and the least value is  −13/12 . Thus the  difference is 76 −  (−1312)  =  2712  =  94