The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

The curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect each other orthogonally, then $$a^2=$$

The surface area of a spherical balloon is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the balloon is , when the radius of the balloon is $$6 \mathrm{~cm}$$.

If $$f(x)=2x^3-15x^2-144x-7$$, then $$f(x)$$ is strictly decreasing in