The function $$f(x)=\frac{x}{2}+\frac{2}{x}$$ has a local minimum at

$$ f(x)=2 x-\tan ^{-1} x-\log (x+\sqrt{x^2+1}) \text { is monotonically increasing, when } $$

The altitude of a cone is $$20 \mathrm{~cm}$$ and its semi vertical angle is $$30^{\circ}$$. If the semi vertical angle is increasing at the rate of $$2^0$$ per second, then the radius of the base is increasing at the rate of

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is