The triangle of maximum area that can be inscribed in a given circle of radius $r$ is ...... .

$A$ circular sector of perimeter $60 \ m$ with maximum area is to be constructed. The radius of the circular arc in meters must be: (in $m$)

Let $f(x) = x^2 + \frac{1}{x^2}$ and $g(x) = x - \frac{1}{x}$,$x \in R - \{-1, 1, 0\}$. If $h(x) = \frac{f(x)}{g(x)}$,then the local minimum value of $h(x)$ is:

$A$ solid hemisphere is mounted on a solid cylinder,both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume,then the ratio of the height of the cylinder to the common radius is

Let $f_1:(0, \infty) \rightarrow \mathbb{R}$ and $f_2:(0, \infty) \rightarrow \mathbb{R}$ be defined by
$f_1(x) = \int_0^x \prod_{j=1}^{21}(t - j)^j dt, x > 0$
and
$f_2(x) = 2(x-1)^{50} - 25(x-1)^{48} + 2450, x > 0,$
where,for any positive integer $n$ and real numbers $a_1, a_2, \ldots, a_n$,$\prod_{i=1}^n a_i$ denotes the product of $a_1, a_2, \ldots, a_n$. Let $m_i$ and $n_i$,respectively,denote the number of points of local minima and the number of points of local maxima of function $f_i, i=1, 2$,in the interval $(0, \infty)$.
$(1)$ The value of $2m_1 + 3n_1 + m_1n_1$ is.
$(2)$ The value of $6m_2 + 4n_2 + 8m_2n_2$ is.
Find the values for $(1)$ and $(2)$.

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: