The difference between the greatest and least values of the function $f\left(x\right)=\cos x+\frac{1}{2}\cos 2x-\frac{1}{3}\cos 3xis$

detailed solution

Correct option is C

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