$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $
$$y=2\cos A\cos B\cos C=[\cos(A-B)+\cos(A+B)]\cos C=[\cos(A-B)-\cos C]\cos C$$
$$\implies\cos^2C-\cos(A-B)\cos C+y=0$$ which is Quadratic Equation in $\cos C$
As $C$ is real $\implies\cos C$ is real,
the discriminant $\cos^2(A-B)-4y\ge0\iff y\le\dfrac{\cos^2(A-B)}4\le\dfrac14$
Homage a lab
$2\cos A\cos B\cos C=(\cos(A+B)+\cos(A-B))\cos C=(\cos(A-B)-\cos C)\cos C=\frac{1}{4}\cos^2(A-B)-(\cos C-\frac{1}{2}\cos(A-B))^2\le\frac{1}{4}$