The maximum value of the function $f(x) = 3x^3 - 18x^2 + 27x - 40$ on the set $S = \{x \in R : x^2 + 30 \leq 11x\}$ is

If a right circular cone having maximum volume is inscribed in a sphere of radius $3 \, cm$,then the curved surface area (in $cm^2$) of this cone is

Let a function $f(x)$ be defined as $f(x) = \begin{cases} \cos^{-1}(\mu) + x^2, & 0 < x < 1 \\ 4x, & x \geqslant 1 \end{cases}$. The function $f(x)$ can have a local minimum at $x = 1$ if the value of $\mu$ lies in the interval:

The cost of running a bus from $A$ to $B$ is $Rs. \left( av + \frac{b}{v} \right)$,where $v \text{ km/h}$ is the average speed of the bus. When the bus travels at $30 \text{ km/h}$,the cost is $Rs. 75$,while at $40 \text{ km/h}$,it is $Rs. 65$. The most economical speed (in $\text{km/h}$) of the bus is:

The function $f(x)=x^3+a x^2+b x+c$ with $a^2 \leq 3 b$ has:

Show that the 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 the cylinder is $\frac{4}{27} \pi h^{3} \tan^{2} \alpha$.