The points on the curve $y = 12x - x^3$ at which the gradient is zero are

The vertices of a triangle are $(0, 0)$,$(x, \cos x)$,and $(\sin^3 x, 0)$ where $0 < x < \frac{\pi}{2}$. The maximum area for such a triangle in sq. units is:

The maximum value of $\frac{\log x}{x}, 0 < x < \infty$ is

$A$ stone is thrown vertically upwards and the height $x \text{ ft}$ reached by the stone in $t$ seconds is given by $x = 80t - 16t^2$. The stone reaches the maximum height in (in $\text{ s}$)

$A$ square piece of tin of side $30\,cm$ is to be made into a box without a top by cutting a square of side $x$ from each corner and folding up the flaps to form a box. If the volume of the box is maximum,then its surface area (in $cm^2$) is equal to $............$.

The point on the curve $x^2 = 2y$ which is nearest to the point $(0, 5)$ is . . . . . . .