Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in $sq.m$) of the flowerbed is

$A$ wire $40 \text{ m}$ in length is to be cut into two pieces. One piece is formed into a square and the other piece into a circle. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum?

Let $AP$ and $BQ$ be two vertical poles at points $A$ and $B$,respectively. If $AP=16 \, m, BQ=22 \, m$ and $AB=20 \, m,$ then find the distance of a point $R$ on $AB$ from the point $A$ such that $RP^2 + RQ^2$ is minimum. (in $, m$)

Suppose $x_1$ and $x_2$ are the point of maximum and the point of minimum respectively of the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$. If the equality $x_1^2 = x_2$ holds true,then the value of $a$ must be:

For the function $f(x) = x^{40} - x^{20}$,find the absolute minimum value in the interval $[0, 1]$.

Find the total number of local maxima and local minima of the function $f(x) = \begin{cases} (2+x)^3, & -3 < x \leq -1 \\ x^{2/3}, & -1 < x < 2 \end{cases}$