$A$ wire of length $20$ units is divided into two parts such that the product of one part and the cube of the other part is maximum. Then,the product of these parts is:

The curve $y(x) = ax^{3} + bx^{2} + cx + 5$ touches the $x$-axis at the point $P(-2, 0)$ and cuts the $y$-axis at the point $Q$,where the derivative $y'(0) = 3$. Find the local maximum value of $y(x)$.

The height of the cone of maximum volume inscribed in a sphere of radius $R$ is

The ordinates of the points on the curve $y = \tan^{-1}(\sin \sqrt{x})$,$0 \leq x \leq 8\pi^2$,at which the tangent is parallel to the $x$-axis are

All the points on the curve $y^{2}=4a|x+a \sin(x/a)|$,where the tangent is parallel to the $x$-axis,lie on:

The function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ where $a > 0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2 = q$, then $a =$