If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$ where $a>0$ attains its maximum and minimum at $p$ and $q$ respectively such that $p^{2}=q$, then $a$ is equal to

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta > 0$ be a very small real number. Let $c \in (a, b)$ be such that $f(c - \delta) < f(c)$ and $f(c + \delta) < f(c)$ for every $\delta > 0$. Let $(f(\alpha - \delta) - f(\alpha))(f(\alpha + \delta) - f(\alpha)) < 0$ for all $\alpha \in (a, b)$ and $\alpha \neq c$. Then:

The maximum value of $f(x) = x^2 \log x$ on the interval $[1, e]$ is:

Find the semi-vertical angle of a right circular cone of a given slant height,if the volume of the cone is maximum.

$A$ sector is removed from a metallic disc and the remaining region is bent into the shape of a circular conical funnel with volume $2 \sqrt{3} \pi$. The least possible diameter of the disc is

The ratio of the altitude of the cone of greatest volume which can be inscribed in a given sphere to the diameter of the sphere is: