$A$ 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:

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 $f(x) = \sin x - x \cos x$ has a maximum at $x = n\pi$,then which of the following is true?

Let $x=2$ be a local minima of the function $f(x)=2x^4-18x^2+8x+12$,$x \in (-4,4)$. If $M$ is the local maximum value of the function $f$ in $(-4,4)$,then $M =$

If $x$ and $y$ are two positive integers such that $x + 2y = 10$ and $x^2 y^3$ is maximum,then $x^2 + 2y^3 =$

The sum of two non-zero numbers is $4$. The minimum value of the sum of their reciprocals is