The greatest value of ${\sin }^{3}x+{\cos }^{3}x \text{in }[0,\frac{\pi }{2}]$  is

$f\left(x\right)={\sin }^{3}x+{\cos }^{3}x\Rightarrow f^1\left(x\right)=3{\sin }^{2}x\cos x-3{\cos }^{2}x\sin x=0$

$\Rightarrow \sin x=\cos x\Rightarrow \sin x=0,\cos x=0$

$x=0,x=\frac{\pi }{2},x=\frac{\pi }{4}\Rightarrow$

$$\begin{gathered} Max=max\left\{f\right(0),f\left(\frac{\pi }{2}\right),f\left(\frac{\pi }{4}\right)\} \\ =max\left\{1,1,\frac{1}{\sqrt{2}}\right\} \\ =1 \end{gathered}$$