The minimum value of $2x + 3y,$ when $xy = 6,$ is

If a running track of $500 \ ft$. is to be laid out enclosing a playground,the shape of which is a rectangle with a semicircle at each end,then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

The function $f(x) = x^3 + ax^2 + bx + c$ where $a^2 \leq 3b$ has:

Two sides of a triangle are given. If the area of the triangle is maximum,then the angle between the given sides is (in $^{\circ}$)

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\int_{0}^{x}e^{(x-t)}f(t)dt, x\in R$ and let $g(x)=\int_{0}^{x}(f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x\in R.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to . . . . . . .

Let $f(x) = \begin{cases} |x| & \text{for } 0 < |x| \leqslant 2 \\ 1 & \text{for } x = 0 \end{cases}$. What is the nature of $f$ at $x = 0$?