$\text{ The extreme values of }4\cos \left(x^2\right)\cdot \cos \left(\frac{\pi }{3}+x^2\right)\cdot \cos \left(\frac{\pi }{3}-x^2\right)\text{ are }$

$$\begin{gathered} let x^2=\theta \\ \Rightarrow 4\cos \left(x^2\right)\cdot \cos \left(\frac{\pi }{3}+x^2\right)\cdot \cos \left(\frac{\pi }{3}-x^2\right) \\ =4\cos \theta \cos \left(60°+\theta \right)\cos \left(60°-\theta \right) \\ =4\cos \theta \left({\cos }^{2}60°-{\sin }^2\theta \right) \\ =4\cos \theta \left(\frac{1}{4}-\left(1-{\cos }^2\theta \right)\right) \\ =4\cos \theta \left({\cos }^2\theta -\frac{3}{4}\right) \\ =4{\cos }^3\theta -3\cos \theta \\ =\cos 3\theta \\ Extreme values of \cos 3\theta are -1,1 \end{gathered}$$