$A$ particle is moving in a straight line. At time $t$, the distance of the particle from its starting point is given by $x = t^3 - 6t^2 + t$. Its acceleration will be zero at

The radius of a circular plate is increasing at the rate of $0.01 \text{ cm/sec}$. When the radius is $12 \text{ cm}$,the rate at which the area increases is:

The total cost $C(x)$ in Rupees,associated with the production of $x$ units of an item is given by $C(x) = 0.005x^{3} - 0.02x^{2} + 30x + 5000$. Find the marginal cost when $3$ units are produced,where marginal cost is defined as the instantaneous rate of change of total cost with respect to the number of units produced.

$A$ particle is moving along a line according to the law $S = t^3 - 3t^2 + 4t - 2$,where $S$ is measured in meters and $t$ is measured in seconds. Then the velocity (in $m/s$) of the particle when its acceleration is zero is:

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S = 2t^3 + 2t^2 - 2t - 3$,then the time taken (in seconds) by the particle to change its direction is

The sides of an equilateral triangle are increasing at the rate of $4 \text{ cm/sec}$. Find the rate at which its area is increasing when the side is $14 \text{ cm}$.