Find two positive numbers $x$ and $y$ such that their sum is $35$ and the product $x^{2} y^{5}$ is a maximum.

Statement-$I$: Let the function $f(x) = \begin{cases} -\frac{x}{2} & x < 0 \\ 7x + 8 & x \geq 0 \end{cases}$. Then $f(x)$ has a local minimum at $x = 0$.
Statement-$II$: If $f(a) < f(a - h)$ and $f(a) < f(a + h)$ for a sufficiently small $h > 0$,then $f(x)$ has a local minimum at $x = a$.

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

The minimum value of the function $y = 2x^3 - 21x^2 + 36x - 20$ is

If $20$ is divided into two parts such that the product of the cube of one part and the square of the other part is maximum,then these two parts are:

The minimum value of the slope of the tangent to the curve $y=x^3-3x^2+2x+93$ is