If $f(x)=3x^3-9x^2-27x+15$,then the maximum value of $f(x)$ is $.....$

Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is $\sin ^{-1}\left(\frac{1}{3}\right)$.

The height (in $cm$) of a cylinder of the greatest volume that can be inscribed in a sphere of radius $3 \ cm$ is

$A$ wire of length $64 \text{ m}$ is to be bent to form a rectangle such that its area is maximum. What is the maximum area (in $\text{ m}^2$)?

$A(1,15), B(3,-12), C(6,12)$ are three consecutive turning points of a continuous curve $y=f(x)$. If $f(x)=0$ only for $x=\alpha$ and $x=\beta$,then $|\beta-\alpha| < $

Find the total number of local maxima and local minima of the function $f(x) = \begin{cases} (2+x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$