Determine whether the function are increasing or decreasing for the stated value of $x$ :

$f\left(x\right)=\frac{1}{x}, x<0$.

Determine whether the function are increasing or decreasing for the stated value of $x$ :

$f\left(x\right)=\frac{1}{1+x^2}, x\ge 0$.

Determine whether the function are increasing or decreasing for the stated value of $x$ :

$f\left(x\right)={\cos }^2 x$ in $\left[0, \frac{\mathrm{\pi }}{2}\right]$.

Determine whether the function are increasing or decreasing for the stated value of $x$ :

$f\left(x\right)=-\frac{x}{2}+\sin x, -\frac{\mathrm{\pi }}{3}<x<\frac{\mathrm{\pi }}{3}$.