Miscellaneous Exercise on Chapter 6

Find the intervals in which the function $f$ given by $f\left(x\right)=\frac{4\mathrm{s}\mathrm{i}\mathrm{n}x-2x-x\mathrm{c}\mathrm{o}\mathrm{s}x}{2+\mathrm{c}\mathrm{o}\mathrm{s}x}$ is decreasing.

Find the intervals in which the function $f$ given by $f\left(x\right)=x^3+\frac{1}{x^3},x\ne 0$ is decreasing.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4r}{3}$.

Let $f$ be a function defined on $\left[a,b\right]$ such that $f^{\text{'}}\left(x\right)>0$, for all $x\in \left(a,b\right)$. Then prove that $f$ is an increasing function on $\left(a,b\right)$.

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $\frac{2R}{\sqrt{3}}$. Also find the maximum volume.

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical angle $\alpha$ is one-third that of the cone and the greatest volume of cylinder is $\frac{4}{27}\pi h^3{\tan }^2\alpha$.

A cylindrical tank of radius $10 \mathrm{m}$ is being filled with wheat at the rate of $314$ cubic meter per hour. Then the depth of the wheat is increasing at the rate of

Find the absolute maximum and minimum values of the function $f$ given by $f\left(x\right)=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}x+\mathrm{s}\mathrm{i}\mathrm{n}x,x\in \left[0,\pi \right]$.