The maximum value of $f(x) = \frac{x}{4 + x + x^2}$ on the interval $[-1, 1]$ is:

Let $f: R \rightarrow R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t) dt$ for $x>0$. Which of the following options is/are correct?
$(1)$ $F$ has a local minimum at $x=1$
$(2)$ $F$ has a local maximum at $x=2$
$(3)$ $F(x) \neq 0$ for all $x \in (0,5)$
$(4)$ $F$ has two local maxima and one local minimum in $(0, \infty)$

Find the semi-vertical angle of a right circular cone of a given slant height,if the volume of the cone is maximum.

An open tank with a square bottom is to contain $4000 \ cm^3$ of liquid. Find the dimensions of the tank such that the surface area of the tank is minimum.

The equations of motion of two stones thrown vertically upwards simultaneously are $s_1 = 19.6t - 4.9t^2$ and $s_2 = 9.8t - 4.9t^2$ respectively. If the maximum height attained by the first stone is $h$,then when the first stone is at its maximum height,the height of the second stone will be:

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