For all real $x$,the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is

Let $f(x) = x^3 + px + 1$ and consider the following three statements:
$(i)$ For $p \geqslant 0$,$f(x) = 0$ has one negative root and $f(x)$ is monotonic.
$(ii)$ For $-1 < p < 0$,$f(x) = 0$ has one negative root and $f(x)$ is non-monotonic.
$(iii)$ For $p < -3/\sqrt[3]{4}$,$f(x) = 0$ has three real and distinct roots.
Which of the following is correct?

Suppose $x_1$ and $x_2$ are the point of maximum and the point of minimum respectively of the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If the equality $x_1^2 = x_2$ holds true,then the value of $a$ must be:

The minimum value of $\left( x^2 + \frac{250}{x} \right)$ is

If the sum of two numbers is $3$,then the maximum value of the product of the first number and the square of the second number 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 $(2)$.