Two particles $P$ and $Q$ located at the points $P(t, t^3 - 16t - 3)$ and $Q(t + 1, t^3 - 6t - 6)$ are moving in a plane. The minimum distance between the points during their motion is:

$A$ box open from the top is made from a rectangular sheet of dimensions $a \times b$ by cutting squares of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum,then $x$ is equal to:

If the local maximum value of the function $f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^2 x}, \quad x \in\left(0, \frac{\pi}{2}\right)$,is $\frac{k}{e}$,then $\left(\frac{ k }{ e }\right)^8+\frac{ k ^8}{ e ^5}+ k ^8$ is equal to

$A$ rectangular sheet of tin $45 \, cm$ by $24 \, cm$ is to be made into a box without a top by cutting off a square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum (in $, cm$)?

The number of turning points of the curve $f(x) = 2 \cos x - \sin 2x$ in the interval $[-\pi, \pi]$ is

The difference between the local extreme values of the function $f(x) = 2x^3 - 15x^2 + 36x + 40$ is ......