The minimum value of $\frac{(5 + x)(2 + x)}{1 + x}$ for non-negative real $x$ 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?

If $f(x)=p x^3+q x^2+r x+t$ attains local minimum and local maximum values at $x=-2$ and $x=2$ respectively and $p$ is a root of $9 x^2-1=0$,then $p+q+r=$

The minimum distance between a point on the curve $y=e^x$ and a point on the curve $y=\log_e x$ is

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

The absolute maximum value of the function $f(x)=2 x^3-9 x^2+12 x+1$ on the interval $[0,2]$ is: