The maximum value of $(1/x)^x$ is -

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 . . . . . . .

Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be,if the area of the flower bed is to be the greatest (in $m$)?

$A$ 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 function $f(x) = x + \sin x$ has

The minimum value of the function $f(x)=2|x-1|+|x-2|$ is