The minimum value of $e^{(2x^2 - 2x + 1)\sin^2 x}$ is

$A$ 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 $, cm$)?

Let $M$ and $m$ be respectively the local maximum and the local minimum values of the function $f(x) = 2x^3 - 9x^2 + 12x + 5$ in the interval $[0, 3]$. Then $M - m$ is equal to

The maximum value of the function $f(x) = -|x+1| + 3, x \in R$ is . . . . . . .

The acceleration $f \text{ ft/sec}^2$ of a particle after a time $t \text{ sec}$ starting from rest is given by $f = 6 - \sqrt{1.2t}$. Then the maximum velocity $v$ and time $T$ to attain this velocity are:

Consider all rectangles lying in the region $\left\{( x , y ) \in R \times R : 0 \leq x \leq \frac{\pi}{2} \text{ and } 0 \leq y \leq 2 \sin (2 x )\right\}$ and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles is