The maximum volume of the right circular cone with slant height $6$ units is

On the interval $[1, e]$,the greatest value of $f(x) = x^2 \log x$ is:

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

If $20$ is divided into two parts such that the product of the cube of one part and the square of the other part is maximum,then these two parts are:

If $f(x) = \int_{x}^{x^2} (t - 1) \, dt$ for $1 \le x \le 3$,then the global maximum value of $f(x)$ is

$A$ rectangle with one side lying along the x-axis is to be inscribed in the closed region of the $xy$ plane bounded by the lines $y = 0$,$y = 3x$,and $y = 30 - 2x$. The largest area of such a rectangle is