Find the maximum and minimum values of the function given by $h(x) = x + 1, x \in (-1, 1)$.
(NONE) The given function is $h(x) = x + 1$ defined on the open interval $(-1, 1)$.
For any $x \in (-1, 1)$,we have $-1 < x < 1$.
Adding $1$ to all parts of the inequality,we get $-1 + 1 < x + 1 < 1 + 1$,which simplifies to $0 < h(x) < 2$.
As $x$ approaches $-1$ from the right,$h(x)$ approaches $0$,but $h(x)$ is never equal to $0$ because $-1$ is not included in the domain.
As $x$ approaches $1$ from the left,$h(x)$ approaches $2$,but $h(x)$ is never equal to $2$ because $1$ is not included in the domain.
Since the function is strictly increasing and the interval is open,there is no point $c \in (-1, 1)$ such that $h(c)$ is the maximum or minimum value.
Therefore,the function $h(x)$ has neither a maximum nor a minimum value in the interval $(-1, 1)$.
For any $x \in (-1, 1)$,we have $-1 < x < 1$.
Adding $1$ to all parts of the inequality,we get $-1 + 1 < x + 1 < 1 + 1$,which simplifies to $0 < h(x) < 2$.
As $x$ approaches $-1$ from the right,$h(x)$ approaches $0$,but $h(x)$ is never equal to $0$ because $-1$ is not included in the domain.
As $x$ approaches $1$ from the left,$h(x)$ approaches $2$,but $h(x)$ is never equal to $2$ because $1$ is not included in the domain.
Since the function is strictly increasing and the interval is open,there is no point $c \in (-1, 1)$ such that $h(c)$ is the maximum or minimum value.
Therefore,the function $h(x)$ has neither a maximum nor a minimum value in the interval $(-1, 1)$.
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$f_1(x) = \int_0^x \prod_{j=1}^{21}(t - j)^j dt, x > 0$
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 $(2)$.
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