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

The maximum value of $f(x) = \frac{\log x}{x}$ $(x > 0, x \neq 1)$ is

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum, then the numbers are:

The function $f(x) = x^3 - 6x^2 + 9x + 2$ has a maximum value when $x$ is

Find the absolute maximum value and the absolute minimum value of the function given by $f(x) = 4x - \frac{1}{2}x^2$ for $x \in \left[-2, \frac{9}{2}\right]$.

The sum of the global minimum and global maximum values of the function $f(x) = \frac{4}{3}x^3 - 4x$ in the interval $[0, 2]$ is: