If $f(x) = 3x + \frac{12}{x}$ is continuous on $R - \{0\}$ and $M$ is its local maximum value,then $\lim_{x \rightarrow M} f(x) = $

Find the absolute maximum and minimum values of the function $f$ given by $f(x) = 12x^{\frac{4}{3}} - 6x^{\frac{1}{3}}$ for $x \in [-1, 1]$.

The maximum value $M$ of $f(x) = 3^x + 5^x - 9^x + 15^x - 25^x$,as $x$ varies over all real numbers,satisfies:

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,find the minimum value of $f(x)$.

$A$ wire of length $20 \ m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in $meters$) of the hexagon,so that the combined area of the square and the hexagon is minimum,is:

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 $\frac{4}{27} \pi h^{3} \tan^{2} \alpha$.