What is the maximum slope of the curve $y = -x^3 + 3x^2 + 9x - 27$?

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

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?

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ is

Find both the maximum value and the minimum value of $3x^{4}-8x^{3}+12x^{2}-48x+25$ on the interval $[0,3]$.

If an open cylinder of given surface area has maximum volume,then its radius is