Maximums are Unequal

Since $a, b, c \in [0,1]$ , we can let $a = \sin^2 \alpha$ , $b = \sin^2 \beta$ and $c = \sin^2 \gamma$ . Then

$\begin{aligned} X = \sqrt{abc} + \sqrt{(1-a)(1-b)(1-c)} & = \sin \alpha \sin \beta \sin \gamma + \cos \alpha \cos \beta \cos \gamma \end{aligned}$

By AM-GM inequality,

$\begin{aligned} X = \sin \alpha \sin \beta \sin \gamma + \cos \alpha \cos \beta \cos \gamma & \le \frac {\sin^3 \alpha + \sin^3 \beta + \sin^3 \gamma}3 + \frac {\cos^3 \alpha + \cos^3 \beta + \cos^3 \gamma}3 \end{aligned}$

And equality occurs when $\alpha = \beta = \gamma = 0$ or $\frac \pi 2$ , and the maximum values of $X$ is $\boxed{1}$ .