The function,$f(x)=x \sqrt{1-x}$,where $x \in(0,1)$,has a local maximum at $x=$

Let $f(x) = x^2 e^{-2x}, x > 0$. The maximum value of $f(x)$ is

Among all sectors of a fixed perimeter,choose the one with maximum area. Then,the angle at the centre of this sector (i.e.,the angle between the bounding radii) is

The difference between the greatest and the least values of the function $f(x) = \sin(2x) - x$ on the interval $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ is:

$A$ particle moving in a straight line starts from rest and the acceleration at any time $t$ is $a - kt^2$,where $a$ and $k$ are positive constants. The maximum velocity attained by the particle is

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha, y=\beta$,then $\frac{k}{\alpha^2 \beta^2} =$