The values of $x$ at the stationary points of $f(x)=x^3+3x^2-2$ are

If $6x - x^2 + 12$ attains its extreme value $\beta$ at $x = \alpha$, then $\beta =$

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 $(2)$.

Find the local maximum and local minimum values for the function given by $g(x) = \frac{x}{2} + \frac{2}{x}, \quad x > 0$.

If local maximum of $f(x) = \frac{ax + b}{(x - 1)(x - 4)}$ exists at $(2, -1)$, then $a + b =$

One maximum point of ${\sin ^p}x{\cos ^q}x$ is